Local Class Field Theory via Lubin-Tate Theory

Local Class Field Theory via Lubin-Tate Theory

by

Adam Mohamed

Thesis presented in partial fulfilment of the requirements for the degree of

Master of Science

at

Stellenbosch University

Supervisor: Dr. Arnold Keet Department of Mathematical Sciences

Faculty of Sciences

Date: December 2008

i

Declaration

By submitting this thesis electronically, I declare that the entirety of thework contained therein is my own, original work, that I am the owner of thecopyright thereof (unless to the extent explicitly otherwise stated) and thatI have not previously in its entirety or in part submitted it for obtaining anyqualification.

Date: November 2008

Copyright c©2008 Stellenbosch University

All rights reserved

ii

Abstract

This is an exposition of the explicit approach to Local Class Field Theorydue to J. Tate and J. Lubin. We mainly follow the treatment given in [15]and [25]. We start with an informal introduction to p-adic numbers. Wethen review the standard theory of valued fields and completion of thosefields. The complete discrete valued fields with finite residue field knownas local fields are our main focus. Number theoretical aspects for localfields are considered. The standard facts about Hensel’s lemma, Galois andramification theory for local fields are treated. This being done, we continueour discussion by introducing the key notion of relative Lubin-Tate formalgroups and modules. The torsion part of a relative Lubin-Tate module isthen used to generate a tower of totally ramified abelian extensions of a localfield. Composing this tower with the maximal unramified extension givesthe maximal abelian extension: this is the local Kronecker-Weber theorem.What remains then is to state and prove the theorems for explicit local classfield theory and end our discussion.

iii

Opsomming

Hierdie tesis is ’n uiteensetting van die eksplisiete beskrywing van klaslig-gaamteorie van J. Tate en J. Lubin. Ons volg die behandeling wat in [15]en [25] gegee is. Ons begin met ’n informele inleiding aan p-adiese getalle.Ons beskou dan die standaarde teorie van liggame met waardering en hulvervollediging. Die volledige diskrete liggame met waardering en ’n eindigeresklasliggaam is bekend as lokaleliggame, en ons fokus op hulle. Ons beskoudie getalleteorie van lokaleliggame. Die bekende feite oor Hensel se lemma,Galois teorie en vertakkingteorie is behandel. Daarna beskou ons die sleutelbegrippe van relatiewe Lubin-Tate formele groepe en modules. Die torsiedeel van ’n relatiewe Lubin-Tate module is gebruik om ’n toring van totalevertakte abelse uitbreidings van ’n lokaleliggaam voort te bring. As ons hi-erdie toring met die maksimum onvertakte uitbreiding saamstel dan kry onsdie maksimale abelse uitbreiding: dit is die lokale Kronecker-Weber stelling.Wat dan oorbly is om die stellings van eksplisiete lokale klasliggaamteoriete stel en te bewys. Ons sluit dan af.

iv

Acknowledgments

This is the place for me to express my deepest gratitude to my supervisor,

Dr. Arnold Keet. His constant support, help, suggestions and teaching have

made this thesis a reality.

I would like also to take this opportunity to thank Prof B. Green and

Prof F. Breuer for all their help.

During my master’s studies and the writing of this thesis I had the

supports of an AIMS partial bursary for master’s studies, Stellenbosch Uni-

versity Science Faculty bursary for graduate studies and I was awarded an

NRF-Africa scholarship for graduate studies. I thank these institutions for

their support.

Contents

1 Introduction 3

2 p-adic Numbers, an Introduction 52.1 Where does all this come from? . . . . . . . . . . . . . . . . . 5

3 Valued and Complete Fields 113.1 Absolute values and Valuations . . . . . . . . . . . . . . . . . 11

3.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 Absolute value or valuation on number fields and ra-

tional function field . . . . . . . . . . . . . . . . . . . 183.2 Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Algebraic extensions of complete valued fields 334.1 Extending absolute values . . . . . . . . . . . . . . . . . . . . 334.2 Galois theory and the norm group of local fields. . . . . . . . 39

4.2.1 Ramification in an extension of a local field. . . . . . . 394.2.2 Galois theoretical aspects for local fields. . . . . . . . 444.2.3 The group of norms . . . . . . . . . . . . . . . . . . . 52

5 Formal group law, Lubin-Tate extensions and Local ClassField Theory 555.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Relative Lubin-Tate formal group law . . . . . . . . . . . . . 56

5.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 565.2.2 Relative Lubin-Tate formal group laws . . . . . . . . . 57

5.3 Relative Lubin-Tate extensions . . . . . . . . . . . . . . . . . 615.3.1 Isomorphism of Lubin-Tate extensions . . . . . . . . . 63

5.4 Local class field theory . . . . . . . . . . . . . . . . . . . . . . 695.4.1 The local Kronecker-Weber theorem . . . . . . . . . . 695.4.2 The theorems of local class field theory . . . . . . . . 71

1

Chapter 1

Introduction

In his famous address at the Second International Conference of Mathe-maticians in Paris in 1900, Hilbert asked among other things whether theKronecker-Weber theorem, that is every abelian extension of Q is containedin some Q(e(2πir)), r ∈ Q, has an analogue for a number field F . This isHilbert 12th Problem or Explicit Class Field Theory. Depending on how Fcan be embedded into C, answers and approaches to this problem elegantlyillustrate the interconnections between the algebraic, analytic and geometricsides of Number Theory.

Indeed in the situation where F is a quadratic imaginary field, the the-ory of Complex Multiplication gives an explicit way to generate all abelianextensions of F, for example the Hilbert class field of F. Here the j invariantof an elliptic curve E/Q with endomorphism ring an order of F , O, playsthe role of e(x).

For a totally real field F, if Starks’s conjectures are true, as has beenproved in many special cases by computation, then Hilbert’s problem wouldhave a positive answer. Here the so-called Stark units play the role of theroots of unity in the classical case. See [19] and [5] for details.

If instead of number fields, i.e., finite extensions of Q, we take finite ex-tensions K of the p-adic numbers Qp, which are local fields, then we have anexplicit class field theory due to J. Lubin and J. Tate. They called it FormalComplex Multiplication in Local Fields. Here we exploit the completenessof K and explicitly construct abelian extensions using a formal group. Themaximal ideal of the algebraic closure of K becomes a module over the ringof integers of K, and we adjoin the torsion points to K to obtain a tower oftotally ramified extensions of K. Composing this with the maximal unram-ified extension of K, we obtain the maximal abelian extension of K. Ouraim in this thesis is to give a nearly self-contained exposition of this theory,following the treatment in [15] and its recent refinement in [25], in whichthe local Kronecker-Weber theorem is proved using the Hasse-Arf theorem.

More precisely we shall prove

3

4 Introduction

Theorem 1.0.1. (Local Class Field Theory ) Let K be a local field. IfK ′/K is an algebraic extension of K, we write N(K ′/K) :=

⋂NL/K(L∗)

where L runs over the finite subextensions of K inside K ′ and let Kur

be the maximal unramified extension of K. Let ϕK be the Frobenius ofGal(Kur/K). Then:

1. There is a unique homomorphism called the Artin map:

ArtK : K∗ → Gal(Kab/K)

characterized by the following properties:• For a prime π ∈ K, then ArtK(π)|Kur = ϕK• For an abelian extension K ′/K, then ArtK(N(K ′/K))|K′ = id.

2. For a finite abelian extension K ′/K, the Artin map induces the exactsequence:

1→ NK′/K(K ′∗)→ K∗ → Gal(K ′/K)→ 1.

To this end we have organized the thesis as follows.In chapter 2 we give an informal introduction to the p-adic numbers Qp.

We will make use of the celebrated Hensel’s lemma to justify that we havestrict a embedding Q ↪→ Qp.

Thereafter, in chapter 3 we review the basic notions of absolute valuesor valuations on a field K and the completion of K. We emphasize non-archimedean absolute values and give a version of Hensel’s lemma for Kcomplete with respect to a non-archimedean absolute value.

In chapter 4 we consider algebraic extensions of a complete discrete valuedfield. In this setting, one important consequence of Hensel’s lemma is theuniqueness of the extension of the absolute value on a complete discretevalued field K to its algebraic extensions. We next define local fields andclassify them. We also discuss some number theoretical aspects on localfields: Galois theory, ramification theory, lower and upper numbering ofhigher ramification groups and we finish with a brief discussion on the normgroup of an extension of a complete discrete valued field.

Lastly, in chapter 5 we introduce the notion of Lubin-Tate formal grouplaw in order to define Lubin-Tate modules. Then torsion points on a Lubin-Tate module give rise to relative Lubin-Tate extensions. These are totallyramified extensions generalizing the construction of adjoining p-power rootsof unity to Qp to obtain totally ramified extensions. Composing the unionof a tower of relative Lubin-Tate extensions with the maximal unramfiedextension build up the abelian closure: this is the local Kronecker-Webertheorem. We end the discussion with a proof of theorem 1.0.1.

Chapter 2

p-adic Numbers, anIntroduction

In this short chapter we explain the idea behind the p-adic number system.Throughout the discussion we will try to shed light on the motivation for thisnew number system and give some consequences of this informal definition.

2.1 Where does all this come from?

The use of p-adic methods to expand an algebraic integer as a sum of powersof a prime appears in Kummer’s work on Fermat Last Theorem and evenbefore. These ad-hoc methods, familiar to working number theorists, havefound a formal setting since Kurt Hensel first viewed them as independentobjects on their own. That is he considered p-adic expansions, i.e., expan-

sions of the form∞∑n=m

anpn where m ∈ Z and an ∈ {1, 2, · · · , p − 1} a set

of representatives of the residue field Z/pZ; independently of the rationalsas such expansions don’t come always from the p-adic expansion of a ra-tional number. We will be more precise on the latter along our informalintroduction to p-adic numbers. p-adic methods in number theory have be-come an important tool since Kurt Hensel and his predecessors have shownthe advantage of applying the methods of series expansions from analysis tonumbers.

Indeed, the p-adic numbers arise when one looks at analogies betweenthe function field C(z) with the field of rational numbers Q. For instance asZ, C[z] is a unique factorization domain. In a standard course on ComplexAnalysis, one studies the Laurent series expansion of f(z) ∈ C(z) around apoint a ∈ C. In some region in C we have the unique expansion

f(z) =∞∑n≥m

cn(z − a)n

5

6 p-adic Numbers, an Introduction

with m ∈ Z and cn ∈ C. This is what is known as local theory in Analysissince we are looking at the behavior of f around a given point a. For instancewe can read if a is a zero or a pole of f and its multiplicity.

Now using the language of algebra, z − a is a prime element in the fieldC(z). Any f(z) ∈ C(z) admits a unique expansion at the prime z− a of theform

f(z) =∞∑n>m

cn(z − a)n,

where the coefficients cn are taken from a set of representative of the residuefield C ∼= C[z]/(z − a).

We can achieve a similar expansion in Q using the analogy:

C(z) ←→ QC[z] ←→ Zz − a ←→ p

C ∼= C[z]/(z − a) ←→ Z/pZ∞∑n>m

cn(z − a)n ←→∞∑n≥l

anpn.

We immediately come across the problem of convergence of such a series.Obviously with respect to the usual absolute value on Q such series don’tconverge. Recall that the real numbers and many other concepts in Mathe-matics were rigorously defined only during the 19-th century. This problemof convergence was the main reason why the mathematicians contemporaryto Kurt Hensel were cautious about this new system of numbers. It wassome years later when the topological tools were ready that the Hungarianmathematician J. Kurschak proposed, by analogy with the construction ofthe real numbers from Q, to view the p-adic numbers as the completion ofthe rational numbers with respect to the p-adic absolute value. This ledto Valuation Theory where topological concepts help one to understand thearithmetic of certain fields. So, let us see how we can expand rational num-bers at a prime number.

We first make the following definition

Definition 2.1.1. The formal set of numbers of the form∑∞

n≥m anpn with

an ∈ {0, 1, · · · , p − 1} and m ∈ Z, is denoted by Qp and called the p-adicnumbers. The subset of these formal numbers of the form a0+a1p+a2p

2+· · ·is called the set of p-adic integers and denoted by Zp.

Let a ∈ N and p a prime number. From the Euclidean division we canwrite

a = r0 + a1p

2.1 Where does all this come from? 7

with 0 6 r0 6 p− 1. Do the same with a1, we have

a1 = r1 + a2p

with 0 6 r1 6 p− 1. Thus we obtain

a = r0 + r1p+ a2p2.

with 0 6 r1, r2 6 p − 1. Do the same with a2 and so on. The processterminates, so we can write

a = r0 + r1p+ r2p2 + · · ·+ rlp

l

with ri ∈ {0, 1, 2, · · · , p− 1}.

On trying to do the same process with a negative integer we have toallow an infinite expansion. This is because of the formula

−1 =p− 11− p

=∞∑n=0

(p− 1)pn.

So, for a negative integer a, one has

a = r0 + r1p+ r2p2 + · · ·+ rn−1p

n−1 + (p− 1)pn + (−1)pn+1

= r0 + r1p+ r2p2 + · · ·+ rn−1p

n−1 + (p− 1)pn + (p− 1)pn+1 + · · · .

The next step is to expand a rational number. To do so, we need first todescribe how we can do arithmetic with these numbers. First we need toremark that unlike power series of functions; in the p-adic expansion the setof coefficients is not closed under addition and multiplication. This is similarto arithmetic with real numbers in their decimal expansion. Here is one wayof doing the basic arithmetic operations with p-adic numbers. Without lossof generality we describe the arithmetic with the p-adic integers.

How to add p-adic integers:Take a = a0 + a1p + a2p

2 + a3p3 + · · · , and b = b0 + b1p + b2p

2 + b3p3 +

· · · ,with ai, bi ∈ {0, 1, 2, · · · , p− 1}. We want to find

r = a+ b = (a0 + a1p+ a2p2 + a3p

3 + · · ·) + (b0 + b1p+ b2p2 + b3p

3 + · · ·= a0 + b0 + (a1 + b1)p+ (a1 + b2)p2 + · · · .

But perhaps the ai + bi do not lie in {0, 1, 2, · · · , p− 1}. We write a0 + b0 =c1p+ r0 with 0 6 r 6 p− 1. Thus r becomes

r = r0 + (a1 + b1 + c1)p+ (a2 + b2)p2 + (a2 + b3)p3 + · · · .

Repeat the same process for a1 + b1 + c1, and so on. We see that we have


an algorithm to find the digits of r.How to multiply p-adic integers:

Now we want to give a meaning to m = (a0+a1p+a2p2+· · ·)(b0+b1p+b2p2+

· · ·). We start with m = a0b0 + (a1b0 + b1a0)p+ · · ·+ (∑

i+j=n aibj)pn + · · · .

And as above we write a0b0 = c1p+ r0 with 0 6 r0 6 p− 1, so that we havem = r0 + (a1b0 + b1a0 + c1)p+ · · · . Then write a1b0 + b1a0 + c1 = c2p+ r1;we have in this manner a process to obtain the digits of m. Having definedaddition and multiplication we can now subtract and invert using additionand multiplication: s = a− b = a+ (−1)b, to obtain b = 1

a , we write ba = 1and hence we can find the digits of b from those of a.

We can now represent every rational number r as an expansion in powersof p :

r =∞∑n>m

anpn,

with an ∈ {0, 1, 2, · · · , p− 1} and m an integer. To see that this representa-tion is unique, one introduces the following function.

Definition 2.1.2. For a rational number 0 6= r = pe ab , with p - ab; onedefines vp(r) = e, and vp(0) =∞ with the conventions e+∞ =∞, ∀ e ∈ Z,∞+∞ =∞. This function is called the p-adic valuation.

It satisfies the following properties.

Lemma 2.1.3. Let x, y ∈ Q. Then one has: vp(xy) = vp(x) + vp(y), vp(x+y) ≥ inf{vp(x), vp(y)} with equality if vp(x) 6= vp(y).

Proof. This follows from the definition.

Let r ∈ Q∗. If we write r =∑∞

n≥m anpn where am 6= 0, then vp(r) =

m. Next, if r =∑∞

n≥m anpn =

∑∞i≥j bip

i, with bj 6= 0, it is immediatethat j = m. On the other hand (am − bm)pm +

∑∞n≥m+1(an − bn)pn = 0.

As vp(∑∞

n≥m+1(an − bn)pn) ≥ m + 1, we deduce that vp((am − bm)pm) =∞, i.e., am = bm. Continuing this way , one sees that an = bn for all n ≥m and hence follows the uniqueness of the p-adic expansion of a rationalnumber.

Proposition 2.1.4. With the addition and multiplication given above Zp isan integral domain, and Qp is its quotient field and Q can be embedded inQp.

Proof. This follows from the above discussion.

The valuation vp on Q extends naturally to Qp, and is denoted by vpand called the p-adic valuation.

2.1 Where does all this come from? 9

Remark 2.1.5. By means of vp, we have Zp = {x ∈ Qp : vp(x) ≥ 0}.From vp(x−1) = −vp(x), one deduces that x ∈ Z∗p, the group of units if andonly if vp(x) = 0. Thus the subset m = {x ∈ Zp : vp(x) > 0}, is the onlymaximal ideal of Zp. This has a natural explanation as we shall see in thenext chapter.

Remark 2.1.6. One sees that doing arithmetic with p-adic integers comesdown to modular arithmetic. This is not a coincidence.

With some little work we will be able to see that we have the strictembedding Q ↪→ Qp and that Qp is in fact a “big” field by a countingargument. To spell this out we shall make use of the following result whichis a version of an important result that bears the standard name of Hensel’slemma. A more general version will appear in the sequel.

Proposition 2.1.7. (Hensel’s lemma) Let f(X) =∑d

r=0 crXr ∈ Z[X].

Suppose that there is an integer a such that f(a) ≡ 0 (mod p), f ′(a) 6≡ 0(mod p) with f ′(X) the formal derivative of f(X). Then there exists a uniqueα ∈ Zp such that α ≡ a (mod p) and f(α) = 0.

Proof. The proof consists of finding by induction a sequence of integers{an}n∈N such that f(an) ≡ 0 (mod pn) and an+1 ≡ an (mod pn) whichdetermines uniquely α. For n = 1, set a1 = a. Now, suppose we have founda1, · · · , an satisfying an−1 ≡ an (mod pn−1) and f(an) ≡ 0 (mod pn), andhence the existence of ln ∈ Z with f(an) ≡ lnpn (mod pn+1). Then we wantan+1 such that f(an+1) ≡ 0 (mod pn+1) and an+1 ≡ an (mod pn). Writean+1 = an + knp

n. Hence f(an+1) = f(an + knpn) ≡ f(an) + knp

nf ′(an)(mod p2n) by Taylor expansion for polynomials. Thus f(an) + knp

n ≡ 0(mod pn+1) ⇔ knf

′(an) ≡ −ln (mod p). Since an ≡ a (mod p), we havef ′(an) ≡ f ′(a) 6≡ 0 (mod p), so we can solve for kn and obtain an+1. Toconclude set α = a + k1p + k2p

2 + · · · ∈ Zp. Then α ≡ a (mod p) andf(α) = 0.

Example 2.1.8. Let p be an odd prime number. Take the polynomialf(X) = X2 −m with p - m not a square and is a quadratic residue modulop. Say a2 ≡ m (mod p). Since f ′(a) = 2a 6≡ 0 (mod p), the conditions ofproposition 2.1.7 are satisfied. So there exist sequences {an}n>1 and {kn}n>1

such that an+1 = an + knpn with kn ∈ {0, 1, 2, 3, · · · , p − 1}. Consider the

p-adic integer x = a+k1p+k2p2+· · ·+knpn+· · · . It is a root of f(X) = 0 in

Qp, i.e.,√m ∈ Qp. This shows that for every odd prime p we have Q $ Qp.

Example 2.1.9. For the prime 2, consider the polynomial f(X) = X3−mwith 2 - m a non-perfect cube. So, there is x ∈ Q2 such that x3 −m = 0 inQ2, i.e., 3

√m ∈ Q2.

Therefore for every p we have the strict embedding Q ↪→ Qp. Cantor’sdiagonal argument to prove that R is uncountable can be adapted to provethat the set of p-adic numbers Qp is uncountable.


The real line R is an uncountable field containing Q as dense subfield.The density of Q in Qp is an empty notion since we did not yet define anytopology in Qp. This will be done with a satisfactory construction of Qp

from Q, similarly as R is constructed from Q, by completing Q with respectwith the p-adic absolute value.

Chapter 3

Valued and Complete Fields

In this chapter the formalism that gives a rigorous treatment of the previousdiscussion among other things is introduced. We will start with the basictheory of absolute values and valuations on a field and the completion ofthat field. We will focus on the ultrametric absolute values and give the firstproperties of those fields complete with respect to such an absolute value.The references for this chapter are [1], [9], [23] or [7].

3.1 Absolute values and Valuations

3.1.1 Generalities

By analogy with Cantor’s construction of the real numbers, any valued fieldK can be completed with respect to the metric induced by that absolutevalue.

Let us now turn to the basic theory of absolute values on a field K. Fromthe properties of the usual absolute value C we abstract the definition

Definition 3.1.1. An absolute value on K is a function | . |: K → R>0

satisfying the following axioms

1. |x | = 0⇔ x = 0

2. |xy | = |x || y |

3. |x+ y | 6 |x |+ | y |, this is the triangle inequality.

From the second axiom we deduce that | . | is a homomorphism from themultiplicative group K∗ to the multiplicative group R∗. The subgroup |K∗|of ((R>0)∗, .) is the value group of | . |.

Example 3.1.2. The trivial absolute value has |x | = 1 if 0 6= x.

Thus if K = Fq, the finite field with q elements then every absolute valueon Fq is trivial, because F∗p is torsion and R∗ is torsion free.

11

12 Valued and Complete Fields

Example 3.1.3. In C the usual absolute value is given by |z| = |x + iy| =√x2 + y2. It is the only absolute value on C extending the usual absolute

value on R as we will see in the sequel.

Example 3.1.4. Let K be a number field. If σ : K → C is an embeddingof K, then |x |σ = |σ(x)| is an absolute value on K.

The p-adic absolute value on the rational numbers:We know from commutative algebra that the localization of the integraldomain Z at the prime ideal (p) = pZ is Z(p) = { rs : r ∈ Z , s ∈ Z −pZ}. This ring is principal and it has a unique prime ideal namely pZ(p).A ring satisfying these properties is called a discrete valuation ring. Everya ∈ Z(p) can be written uniquely as a = pn bc with p - bc and n ∈ N.Hence we have a function vp : Z(p) → N ∪ {∞} where one sets for a 6=0, vp(a) = n, and vp(0) =∞. With the usual convention on∞, i.e, n+∞ =∞ and ∞+∞ =∞, it satisfies the following properties

vp(x+ y) > inf{vp(y), vp(x)}vp(xy) = vp(x) + vp(y).

We can extend vp to Q as follows. Any r ∈ Q can be written r = pvp(r) abwith p - ab and a, b, vp(r) ∈ Z. Assign to r ∈ Q its valuation vp(r) at p.This is just the p-adic valuation of r that was defined in chapter 2, p 8.More generally we define

Definition 3.1.5. A valuation on a field K is a function v : K → R ∪ {∞}satisfying the following requirements:

1. v(x) =∞⇔ x = 0

2. v(xy) = v(x) + v(y)

3. v(x+ y) > inf{ v(x), v(y)}.

So v is a homomorphism from (K∗, .) to (R, +). The subgroup v(K∗) of Ris called the value group. A valuation is called discrete when the value groupis isomorphic to Z i.e., v(K∗) = eZ for some 0 6= e ∈ R.

Example 3.1.6. The trivial valuation v : K∗ → R, v(x) = 0 for allx ∈ K∗.

Example 3.1.7. Our prototype of non-trivial valuation is the p-adic valu-ation on Q.

Example 3.1.8. Each x ∈ Q∗p is of the form x =∑∞

n>n0xnp

n with 0 6=n0 ∈ Z. Put v(x) = n0. Then this defines a valuation on Qp. This is theextension of the p-adic valuation of Q to the p-adic field.

3.1 Absolute values and Valuations 13

Example 3.1.9. In the field of formal Laurent series K((X)) in one in-determinate X over a field K, where each f ∈ K((X))∗ is of the formf =

∑∞n>m anX

n, where an ∈ K, am 6= 0, the rule v(f) = m defines avaluation on K((X)).

These examples are instances of discrete valuations as the value groupv(K∗) ∼= Z. If v(K∗) = Z, then one says that the valuation v is normalized.Given a valuation v on a field K, then one can always define an absolutevalue.

Definition 3.1.10. Let 0 < c < 1, then one defines |x |v = cv(x) for x ∈ K.

When K = Q, then the absolute value associated to the p-adic valuationis called p-adic absolute value and is denoted by | . |p. From the propertyv(x+ y) ≥ inf{v(x), v(y)}, one sees that |.|v satisfies:

|x+ y|v ≤ max{|x|v, |y|v}.

This inequality is called the ultrametric inequality . On the real line R theusual absolute value satisfies the Archimedean postulate :

for each x ∈ R, there is n ∈ Z such that |n | > |x |.

In general on a valued field K with absolute value | . |, if for each x ∈ K,there is a ∈ {n.1K : n ∈ Z} such that | a | > |x | then the absolute value iscalled archimedean.This no longer holds with the p-adic absolute value on the rational numbers.Indeed the values |x |p for x ∈ Z are bounded as seen from the ultrametricinequality. Such an absolute value is called non-archimedean. In fact wehave the following characterization of non-archimedean absolute values.

Proposition 3.1.11. Let K be field with char K 6= 0 . Put A = {n.1K :n ∈ Z} the image of Z in K. Then an absolute value | . | on K is non-archimedean if only if it is ultrametric.

Proof. If | . | is ultrametric then

|n.1K | = | 1K + · · ·+ 1K︸︷︷︸n times

| 6 1.

Conversely suppose that there is c such that |x| 6 c for each x ∈ A. We have

|x+ y|l = |(x+ y)l| = |l∑

k=0

(l

k

)xl−kyk|


6l∑

k=0

|(l

k

)||xl−k||yk|

6 cl∑

k=0

|xl−k||yk|

6 cl∑

k=0

(max{|x|, |y|)l

6 c(l + 1)(max{|x|, |y|})l

Then taking l-th root and letting l→∞ give the ultrametric inequality.

For upcoming use let us state

Proposition 3.1.12. Let | . | be an ultrametric absolute value on a field K. If|x | 6= | y |, then |x+ y| = max{|x |, | y |}. Furthermore |x1 + x2 + · · ·+ xn| =max{|x1|, |x2|, · · · , |xn|} if |xi0 | := max{|x1|, |x2|, · · · , |xn|} > |xj | wheni0 6= j.

Proof. The second statement follows by induction from the first. So, ifsay |x | > | y |, then |x | = |x + y − y | ≤ max{|x + y |, | y |}. This shows|x | ≤ |x + y | and from the ultrametric equality we have |x + y | ≤ |x |.Hence |x+ y | = |x |.

As on the real line an absolute value on K defines a metric as follows. If| . | is an absolute value on K, the function d : K×K → R>0 d(x, y) = |x−y|defines a metric on K. Thus the sets B(a, ε) = {x : |a−x| < ε}, ε > 0, forma fundamental system of neighborhoods of a. Hence this defines a topologyon K induced by the metric d.

Definition 3.1.13. Let | . |1, and | . |2 be absolute values on K. They aresaid to be equivalent if their induced topologies are the same.

It is readily seen that this is an equivalence relation. Furthermore wehave

Lemma 3.1.14. If | . |1 and | . |2 are equivalent then |x |1 < 1⇔ |x |2 < 1

Proof. Suppose | . |1 and | . |2 are equivalent. Then

|x |1 < 1⇔ xn → 0 w.r.t | . |1 ⇔ xn → 0 w.r.t | . |2 ⇔ |x |2 < 1.

The converse of lemma 3.1.14 also holds, see the proposition 3.1.17 below.Before we state it, we give two consequences of 3.1.14.


Corollary 3.1.15. An archimedean absolute value cannot be equivalent toa non-archimedean absolute value.

Proof. This is clear.

Corollary 3.1.16. If p and q are distinct primes then the | . |p and | . |q areinequivalent.

Proof. If |x |q = cvq(x) with 0 < c < 1, then | q |q < 1 but | q |p = 1.

The equivalence class of a non-trivial absolute value is determined asfollows.

Proposition 3.1.17. The absolute values equivalent to | . | are exactly theabsolute values | . |s for some positive real number s. In the archimedean casewe have 0 < s ≤ 1.

Proof. When | . |s defines an absolute value then it is equivalent to | . |. Tosee the condition 0 < s ≤ 1 in the archimedean case, write 2s = | 1 + 1 |s ≤| 1 |s + | 1 |s = 2.Conversely suppose that | . |1 is equivalent to | . |2. By the lemma |x |1 < 1⇔|x |2 < 1. Since | . |1 is a non-trivial absolute value we can find a ∈ K∗ suchthat | a |1 < 1. We want to show that | . |1 = | . |s2 for some positive constants. This comes down to showing that the ratio s = log(|x|1)

log(|x|2) is independent of

x ∈ K∗. For that fixed a we want to have log(|a|1)log(|a|2) = log(|x|1)

log(|x|2) ⇔log(|a|1)log(|x|1) =

log(|a|2)log(|x|2) . For any rational number r = m

n with m ∈ Z and n ∈ N we have:

m

n<

log(|a|1)log(|x|1)

⇔ |xma−n|1 < 1

⇔ |xma−n|2 < 1

⇔ m

n<

log(|a|2)log(|x|2)

This implies that log(|a|1)log(|x|1) = log(|a|2)

log(|x|2) . Indeed if we had log(|a|1)log(|x|1) <

log(|a|2)log(|x|2) ,

then there is a rational log(|a|1)log(|x|1) <

mn < log(|a|2)

log(|x|2) which contradicts the above.Hence the result.

Next we shall see that a finite number of inequivalent absolute valuesbehave rather independently from each other.

Proposition 3.1.18. Let | . |1, · · · , | . |n be pairwise inequivalent on K. Thenthere is a sequence {xn}n∈N in K that converges to 1 with respect to | . |1and to 0 with respect to | . |i, i > 2.


Proof. If we can find a ∈ K∗ such that | a |1 > 1 and | a |i < 1, i > 2 thenxn = an

1+an will do. Indeed |xn − 1|1 = | 11+an |1 which converges to 0 and

|xn−0|i = | an1+an |i which converges to 0 for i > 2. The existence of a is givenby the following lemma.

Lemma 3.1.19. With the same hypotheses as above, we can find a ∈ K∗such that | a |1 > 1 and | a |i < 1, i > 2.

Proof. The proof is by induction on n. If | . |1 and | . |2 are inequivalent thenthere is a b ∈ K∗ with |b|1 > 1 and |b|2 6 1. Likewise we have c ∈ K∗ suchthat | c |1 6 1 and | c |2 > 1. Then a = bc−1 is the desired element.Now suppose that we have found b ∈ K∗ with | b |1 > 1 and | b |i < 1, 2 6 i 6n− 1. Since | . |1 and | . |n are inequivalent there exists c ∈ K∗ with | c |1 > 1and | c |n < 1.If | b |n 6 1. Then a = bkc will do for k big enough to ensure that |bkc|i <1, 2 6 i 6 n− 1.Otherwise, | b |n > 1, then form ak = bk

1+bkc.With respect to | . |1 and | . |n, ak

converges to c and with respect to | . |i, 2 6 i 6 n− 1, it converges to 0. Sofor k large enough and by continuity we have the desired a.

Taking d = a−1, we see that we have also found an element in K∗ suchthat | d |1 < 1 and | d |i > 1, 2 6 i 6 n.

The proposition 3.1.18 illustrates the degree of freedom of a finite numberof inequivalent absolute values. This fact does not generalize for infinitelymany inequivalent absolute values as shown by theorem 3.1.30, p 21. Weshall use proposition 3.1.18 to show a result known as the approximationtheorem. This result can be interpreted as a Chinese Remainder Theoremas follows. Take n distinct prime numbers p1, · · · , pn and let x1, · · · , xn beintegers. By the Chinese Remainder Theorem, for every k ∈ N there is aninteger x such that

x ≡ xi (mod pki ).

Let ε > 0, 1pi0

:= maxi{ 1pi}. Choose k an integer such that ( 1

pi0)k ≤ ε. Next

let x ∈ Z be a solution for the congruences x ≡ xi (mod pki ). Then wehave |x − xi |pi ≤ ε. So in terms of the pi-adic absolute values on Q, wemay say that x is as “close” to xi with respect to | . |pi as we please. Thisinterpretation has a weak analogue for K.

Theorem 3.1.20. (The Approximation Theorem) Let | . |1, · · · , | . |n bepairwise inequivalent non-trivial absolute values on a field K. Let x1, · · · , xn ∈K∗. Then for any ε > 0 there exists x ∈ K∗ such that |x− xi|i < ε.

Proof. By proposition 3.1.18 we can find ai arbitrarily close to 1 with respectto | . |i, and arbitrarily close to 0 with respect to | . |j , j 6= i. More preciselyfor any ε > 0 we have | ai − 1 |i < ε

n|xi |i and | ai |j < εn|xi|i , i 6= j. Form


x = x1a1 + · · ·+ xnan, then

|x− xi |i 6 |x1 |i| a1 |i + · · ·+ | ai − 1 |i|xi |i + · · ·+ |xn|i|an|i

<ε|x1|in|x1 |i

+ · · ·+ ε|xn |in|xn |i

= ε.

When dealing with non-archimedean absolute values, one translates thisresult by means of valuations using the correspondence between non-archime-dean absolute values and valuations. One also has the following translationof equivalence for valuations via that correspondence. Two valuations v1and v2 are equivalent if there is a positive real number s such that v1 = sv2.One may also translate the topology induced by an ultrametric absolutevalue in terms of its associated valuation.

In this thesis we will be mainly concerned with the absolute values andvaluations analogous to the p-adic absolute values and the p-adic valuationson Q.

Definition 3.1.21. Let | . | be a non-archimedean absolute value on K andv its associated valuation. Put A| . | = {x ∈ K : |x | 6 1} = {x ∈ K : v(x) >0}. This is called the valuation ring.

Remark 3.1.22. The set A| . | is invariant up to equivalence of valuationsor absolute values.

For simplicity we shall drop the subscript | . | if there is no possible con-fusion. It turns out from the ultrametric property that:

Proposition 3.1.23. A is a ring. The subset m = {x ∈ K : |x | < 1} ={x ∈ K : v(x) > 0} is the only maximal ideal of A. So A is a local ring.

Proof. For x, y ∈ A, then |x + y | 6 max{|x |, |y|} 6 1. Clearly xy ∈ A.Now if x ∈ A−m i.e. |x | = 1. Then |x−1 | = 1. Hence x ∈ A∗ the group ofunits of A. This implies that m is the unique maximal ideal of A.

Definition 3.1.24. The quotient k = A/m which is a field is called theresidue field of (K, | . |).

Note that on a field K endowed with an ultrametric valuation or absolutevalue, the residue field may be infinite. Indeed assume that char(K) 6= 0and consider the field of Laurent series in the variable T over K, K((T ))with the valuation defined in example 3.1.9, p 13. Then the valuation ringis K[[T ]] and its maximal ideal is TK[[T ]] so that its residue field k =K[[T ]]/TK[[T ]] ∼= K. But at the same time we see that if we take K tobe a finite field then the field K((T )) has finite residue field. Those fields


complete with respect to an ultrametric valuation or absolute value are themain object in this thesis. More precisely we will be concerned with thoseof characteristic zero. Before we specialize to this class of fields which areconstructed from number fields as we will see in the sequel, see chapter 4,theorem 4.1.16, p 39; we have to see how one constructs an absolute valueon a number field and rational function field.

3.1.2 Absolute value or valuation on number fields and ra-tional function field

Following the construction of the p-adic absolute values on Q, we constructultrametric absolute values on number fields. To this end let K be a numberfield with ring of integers O . For a non-zero prime ideal p of O we havethe discrete valuation ring Op = {xs : x ∈ O, s ∈ O − p} which is thelocalization of O at p. We know from commutative algebra that the idealpOp = {xs : x ∈ p, s ∈ O − p} is the unique maximal ideal of Op and it isprincipal. Indeed, Op \ pOp = O∗p, the units of Op. This means that up tounits there is only one irreducible element in Op say π, then pOp =< π > .Therefore any element r 6= 0 in Op can be written uniquely as r = πnu withn ∈ N and u ∈ O∗p. Since K is the field of fractions of Op we see that everyelement x 6= 0 of K can be written uniquely as x = πnu with n ∈ Z andu a unit in Op. This defines a valuation vp(x) = n on K. We thus have acorresponding non-archimedean absolute value on K. In fact it turns outthat up to equivalence all non-archimedean absolute values on a number fieldarise this way. To see this, we need the following fact which is interesting inits own right.

Lemma 3.1.25. Let K be a number field with a non-archimedean absolutevalue | . | on it. Let O be the ring of integers of K and let A be the valuationring of K. Then, we have

O ⊆ A.

Proof. Take x ∈ O. Then there are integers ai ∈ Z, 0 6 i 6 n − 1, withn > 1, such that xn + an−1x

n−1 + · · · + a0 = 0. Now suppose that x doesnot lie in A i.e., |x | > 1. Then from proposition 3.1.12 we would have|x | > 1 ⇒ |xn + an−1x

n−1 + · · · + a0 | = |xn | 6= 0 as |xn | > | aixi | fori = 0, · · · , n− 1. A contradiction, and so our assumption is false.

Theorem 3.1.26. There is a 1-1 correspondence between the non-archimedeanabsolute values on K up to equivalence and the non-zero prime ideals of O.

Proof. Given a non-zero prime ideal p of O we have the p-adic absolutevalue of K. Let p and q be two non-zero distinct prime ideals of O. Then,p + q = O, so that we have x ∈ p and y ∈ q such that x + y = 1. Thismeans |x|p < 1 and |y|q < 1. If these absolute values were equivalent, then|1 − y|q < 1, hence it follows that 1 − y ∈ q which gives the contradiction


1 ∈ q. This shows injectivity. Now let | . | be a representative for a classof equivalent non-archimedean absolute values on K. The valuation ring Aassociated to | . | with maximal ideal m contains the ring of integers O of K.So, O∩m = p is a non-zero prime ideal of O. By definition of m, | . | is trivialon O\ p. Hence in the localized ring Op the units O∗p have absolute value 1.As the prime ideal pOp is principal, there is π ∈ Op such that pOp = πOp.Hence every element x ∈ K can be uniquely written as uπk with k ∈ Z andu ∈ O∗p. Therefore |x | = |π |k = |x |p.

In contrast with the above result, the proposition 3.2.9, p 25, shows thatthere are only finitely many archimedean absolute values on K. Here is howthey are constructed.

Let K be a number field, there are n = [K : Q] embeddings γ : K −→ C.If γ(K) ⊆ R, then γ is called real, otherwise γ is called complex. Thecomplex embeddings come in pairs, γ, γ with γ(x) = γ(x) ( the bar denotesthe complex conjugation in C). So if r is the number of real embeddingsand s is the number of pairs of complex embeddings, then n = r + 2s. Nowlet | . | be the usual absolute value on C.For every embedding γ of K in C define |x |γ = | γ(x) |. It is clear that |x |γdefines an archimedean absolute value, and furthermore | . |γ = | . |γ . Up toconjugation we will see that the | . |γ are inequivalent absolute values. Hencewe have constructed n− s inequivalent absolute values on K where n is thenumber of embeddings of K and s the number of pairs of the complex ones.We will show that any archimedean absolute value on K is equivalent to oneof these. So this will classify all the absolute values on a number field K.For K = Q, this result was first established by S. Ostrowsky.

Theorem 3.1.27. ( Ostrowsky)Any absolute value ψ on Q is either equiv-alent to a p-adic absolute value or to the usual absolute value.

Proof. If ψ is non-archimedean, then the above theorem 3.1.26 applies andwe are done in this case.Now suppose that ψ is archimedean. As every rational number is a quotientof integers, ψ(−1) = 1 and ψ is multiplicative, it is enough to show thatfor n ∈ N, ψ(n) = na with 0 < a ≤ 1. That is to say N 3 n 7→ log(ψ(n))

log(n)is a constant function. By the triangle inequality, every n ∈ N satisfiesψ(n) ≤ n. Hence ψ(n) = nb with b a positive real number so that ψ(n) ≥ 1.Let n, m ∈ N arbitrary. We can write any integer power of m as

ml =r∑i=0

aini

with ai ∈ {0, · · · n−1}, ar 6= 0 and r is the integer part of log(ml)/ log(n) =


l log(m)/ log(n). This gives r/l ≤ log(m)/ log(n). We then have

ψ(m)l ≤r∑i=0

ψ(ai)ψ(n)i

≤r∑i=0

nψ(n)i (since ψ(ai) < n)

≤ (r + 1)n (max{1, ψ(n)r}) (because ψ(n)i

≤ max{1, ψ(n)r} for 0 ≤ i ≤ r)= (r + 1)n(max{1, ψ(n)})r

=⇒ ψ(m) ≤ {(r + 1)n}1l (max{1, ψ(n)})

rl

≤ {(r + 1)n}1l (max{1, ψ(n)})

log(m)log(n) .

As ψ(n) ≥ 1 and let l→∞, we obtain that

ψ(m)1

log(m) ≤ ψ(n)1

log(n) .

Since the roles of m and n are symmetric we see that ψ(m)1

log(m) = ψ(n)1

log(n)

with m, n arbitrary integers. Hence log(ψ(n))log(n) =: a is a constant ∈ (0, 1] so

that ψ(n) = na. This is what we wanted.

Having this in hand, we shall now consider instead of Q the rationalfunction field in one variable over a field F , F (X). This shows one aspect ofthe analogy between F (X) and Q. Let f ∈ F (X), f = h

g with g, h ∈ F [X].Define the degree of f to be deg(f) = deg(h) − deg(g) ∈ Z with the usualconvention deg(0) = −∞.

Definition 3.1.28. Take a real number a > 1. Define the degree absolutevalue | . |deg as:

| . |∞ : F (X) −→ R≥0

f 7→ adeg(f)

This is indeed an absolute which is trivial on F and hence non-archimedean.We next associate to each irreducible polynomial p in F [X] an absolute

value as we did for the rational primes. Let p be irreducible in F [X] andf = h

g ∈ F (X). Then we can write f = pvp(f) h1g1

with p coprime withh1g1 and vp(f) ∈ Z. Now let c be real a number such that 0 < c < 1, put| f |p := cvp(f). Similarly as for the p-adic absolute value on Q, one verifiesthat | . |p is a non-archimedean absolute value independent of the choice of c.Hence we have defined infinitely many inequivalent absolute values comingfrom the irreducible polynomials p ∈ F [X]. The next result shows that wehave an exhaustive list of the absolute values on F (X) trivial on F.


Theorem 3.1.29. We fix the rational function field F (X). Then a non-trivial absolute value | . | on F (X) which trivial on F is either equivalent to| . |∞ or | . |p for some irreducible polynomial p ∈ F [X].

Proof. By assumption | . | is non-archimedean. It is enough to consider | f |for f a non-constant polynomial in the ring F [X]. So, let f = anX

n +· · · + a0 ∈ F [X] with an 6= 0. By the ultrametric inequality we have | f | ≤max{|X |n, · · · , | a0 |}. Therefore if |X | > 1, we obtain that | f | = |X |n.Thus | . | is equivalent to | . |∞ in this case.Now if |X | ≤ 1, F [X] lies inside the valuation ring A = {f ∈ F (X) : | f | ≤1}. And one concludes similarly to the number field case.

Next we consider all possible absolute values on Q and on a rationalfunction field F (X) and state a result that gives a relation between them.This result is known as the product formula and is obtained after normalizingthe absolute values on these prime fields. Before stating it, we shall mentionthat the product formula holds in a more general context. That is to saythat this important formula holds when instead of Q or F (X) one has anumber field or a function field. In this context, the product formula is aconsequence of the one given below , for the details see [18, pp 184-185].The normalization of absolute values is done as follows.

In the general setting of a number field K, one has a natural choice forthe base a of a non-archimedean absolute value | . |p on K. Indeed, the primeideal p has a finite residue field OK/pOK . Hence the base a := q−1 whereq = card(OK/pOK). When K = Q, then the residue field is the finite fieldwith p elements, so that |x |p = p−vp(x) for x ∈ Q. For an archimedeanabsolute | . | value on K, we know that |x | = |σ(x) | with σ : K → C, anembedding. Then one puts |x |1 = |x | if σ is real or |x |1 = |x |2 otherwise.Note that the latter is not an absolute value, but this is needed in order toobtain the product formula in general context, see again [18, pp 184-185].The resulting absolute values are called normalized absolute values.

In the context of a rational function field F (X), the absolute value as-sociated to an irreducible polynomial p is normalized as follows. We fixc ∈ (0, 1), and put |x |p = cdeg(p)vp(x) and | f |∞ = (c−1)deg(f).The importance of normalization of absolute values lies in what follows.

Theorem 3.1.30. (Product Formula) Let 0 6= x ∈ F (X), or Q. Then∏|x | = 1,

where the product runs over the normalized archimedean absolute value andthe non-archimedean absolute values in the case of Q and in the case of F (X)the product runs over the normalized non-archimedean absolute values thatare defined above.


Proof. We first have to check that this infinite product is well defined in bothcases. Note that only a finite number of primes or irreducible polynomialsoccurs in the factorization of x. This says that almost all the factors inthe infinite product are 1. The normalization is chosen so that the formulaholds.

After having classified the absolute values and valuations on the mostrelevant fields in number theory, namely number fields and rational functionfields, we shall now turn to the important concept of completion.

3.2 Complete Fields

As a prototype for completing fields one takes the example of the construc-tion of the real numbers from the rationals by using the usual absolute value| . |∞. Here the notion of Cauchy sequence plays a fundamental role. We shalldo it for a general valued field K.

Definition 3.2.1. Let | . | be an absolute value onK and {an}n∈N a sequencein K. Then {an} is called a Cauchy sequence if for every ε > 0 there is l ∈ Nsuch that if m > n ≥ l then | am − an | < ε.

There is of course an equivalent definition of Cauchy sequence by meansof a valuation defined on K. This reads as follows. Let v : K −→ Z ∪ {∞}be a valuation and {an}n∈N a sequence in K. Then {an}n∈N is Cauchy withrespect to v if:

∀m ∈ N, ∃N ∈ N, ∀n ≥ N, ∀ l ∈ N; v(an+l − an) ≥ m.

In other words in a Cauchy sequence the terms become arbitrarily closewith respect to an absolute value from a certain step. So, one would expectCauchy sequences to converge to a limit in their underlying field with respectto the metric topology coming from the absolute value. If this is the casethe field is said to be complete, otherwise it is incomplete. Every field iscomplete with respect to the trivial absolute values. In general with respectto a non-archimedean absolute value, the values of the terms in a Cauchysequence which does not converge to zero become stationary at a certainstep. Indeed we have the proposition.

Proposition 3.2.2. Let K be a field equipped with an ultrametric absolutevalue | . | and let {xn} be a Cauchy sequence not converging to zero. Thenthere exists m ∈ N such that for all n ≥ m we have |xn | = |xm |.

Proof. Since {xn} is a Cauchy sequence not converging to zero, we can finda positive real number δ such that for infinitely many n we have |xn | ≥ δ.This means that the sequence of positive real numbers {|xn |} which isCauchy has a limit greater or equal to δ. Therefore there is N ∈ N such

3.2 Complete Fields 23

that for all n ≥ N we have |xn | > 23δ and |xn+p − xn | < δ

2 for all p ∈ Nso that |xn+p − xn | < |xn |. Now from the ultrametric inequality we have|xn+p | = |xn+p− xn + xn | = max{|xn+p− xn |, |xn |} = |xn | for all p ∈ Nand n ≥ N. This ends the proof.

The rational numbers endowed with the usual topology are incompleteand its completion is just the usual field of real numbers. This is a “wellknown” fact, so, we shall not discuss it here. We shall instead show that therational numbers are also incomplete with respect to a p-adic topology.

We fix a rational prime p and as usual | . |p is the p-adic absolute value onQ. We also choose a polynomial f in Z[X] satisfying the following properties:

1. f is irreducible in Q[X] of degree ≥ 2

2. There is s ∈ Z such that f(s) ≡ 0 (mod p)

3. f ′(s) 6≡ 0 (mod p) where f ′ is the formal derivative of f.

By the Hensel’s lemma, proposition 2.1.7, there is a sequence of integers{an}n∈N such that f(an) ≡ 0 (mod pn) and an+1 ≡ an (mod pn). Thesequence {an}n∈N is a Cauchy sequence in Q with respect to the p-adictopology and converges to an irrational (not in Q) element in Qp. Indeed,on one hand for m > n we have | am − an|p 6 p−n, which shows that it isCauchy. Thus p-adically the sequence {an}n∈N converges to a. But, a is notrational by 1. This establishes the incompleteness of the rational numberswith respect to the p-adic absolute value.

The general setting when we are given a field K with absolute value| . |, we can construct the completion of K by imitating Cantor’s methodof construction of the real numbers R from the rational numbers Q. Tothis end we set C the set of all Cauchy sequences with entries in K. For{an}, {bn} ∈ C define {an}+ {bn} = {an + bn} and {an} × {bn} = {anbn}.

Lemma 3.2.3. With these operations C is a commutative ring with unitelement the sequence {1}n∈N and the zero element {0}n∈N.

Proof. By the the triangle inequality, if {an}, {bn} ∈ C, then {an + bn} ∈ Cas well. From the identity ambm − anbn = an(bm − bn) + bm(am − an) weobtain | ambm − anbn | ≤ | an || bm − bn |+ | bm || am − an |. This implies thatfor {an}, {bn} ∈ C we have {anbn} ∈ C. Indeed, {| an |}, {| bn |} have upperbounds as they are Cauchy.

Consider the subset N ⊂ C of all the sequences that converge to zero inK called the set of null sequences.

Lemma 3.2.4. N is the only maximal ideal of the ring C.


Proof. It is easy to check that N is an ideal. We check the maximality ofN . For {bn} ∈ C \N , from a certain step l the terms of a non-null sequenceare non-zero. So, set cn = 0 for n < l and cn = b−1

n otherwise. Then thesequence {cn} is Cauchy and {bncn} converges to 1. This shows that {bn} isa unit of C and N is indeed the only maximal ideal of C.

Now, we can prove:

Theorem 3.2.5. Let K be a valued field with absolute value | |. Then thereis a field K with the following properties:

1. There is an embedding φ : K ↪→ K; hence K is an extension of K.

2. There is an absolute | . |K on K that extends | . |.

3. In the topology induced by | . |K K is dense in K and K is complete

4. K is unique up to topological isomorphism.

Proof. Existence: We put K = C/N . For each a ∈ K the stationary se-quence {a}n∈N is Cauchy and so we have φ : K ↪→ K. Define | . |K : K −→R≥0, α ≡ {an} (mod N ) 7−→ |α |K = lim

n→∞| an |. First note that since {an}

is Cauchy then {| an |} is Cauchy in the complete field R, hence limn→∞

| an | iswell defined. It is clear that if α ≡ β (mod N ) then |α |K = |β |K . All theother axioms follow easily and | . |K restricted to K is | . |.Density:Take α ≡ {an} (mod N ). We identify K with its image in K, wehave by definition | an − α|K = lim

l→∞| al − an |. Since {an} is Cauchy we see

that the sequence {an} ∈ K converges to α ∈ K.Completeness: Suppose that {αn}n∈N is a Cauchy sequence in K. For eachαn let {an,k}k∈N be a Cauchy sequence in K with αn ≡ {an,k} (mod N ).Consider the sequence a1,1, a2,2, · · · , an,n, · · · . The sequence {an,n} is Cauchy.Indeed, since limk→∞ an,k = αn, ∃l ∈ N, ∀k ≥ l, | an,k − αn|K ≤ min{ 1

n ,1k}.

Therefore, | an,n−am,m |K = | an.n−am,m | ≤ | an,n−αa |K+ | am,m−αm|K+|αm − αm |. So, let α be the image of {an,n}n∈N in K, hence |α − αn |K ≤|α− an,n |K + | an,n − αn|K . The completeness of K is established.Uniqueness: Suppose (F, | . |F ) ⊃ (K, | . |) with the above properties.Then we define a surjective ring homomorphism ψ: C → F, ψ(α) = lim

n→∞an

where α ≡ {an} (mod N ). This is continuous by definition. Therefore wehave a continuous isomorphism K = C/N . Suppose there is another con-tinuous isomorphism φ : K → F. Then φ = ψ on K, and hence we haveequality on K since K is dense in K.

Example 3.2.6. The completion of Q(i) with respect to the usual absolutevalue on the complex numbers is R(i) = C.


Example 3.2.7. We shall soon see that the completion of the rational num-bers that with respect the p-adic absolute values is the p-adic field Qp.

Given an absolute value | . | on a field K, we can characterize the com-pletion of K with respect to | . |. To start with suppose the absolute value isarchimedean. NecessarilyK is of characteristic zero as there is no archimedeanabsolute value on a field of positive characteristic. Hence, we have Q ⊆ K.Let F be the completion of K with respect to | . |. We therefore have R ⊆ F.The next theorem says that in the case of strict inclusion, then, F is the fieldof complex number C. This fundamental fact was first proved by Ostrowsky.

Theorem 3.2.8. (Ostrowsky). The only complete archimedean fields arethe real numbers R and the complex numbers C.

Proof. See [1, p 24], [18, p 124], or [23, p 13].

Ostrowsky’s theorem has the following important consequence for archime-dean absolute values on a number field.

Proposition 3.2.9. Let K be a number field. To an embedding σ : K →C corresponds an absolute value |x |σ = |σ(x) | on K. This induces a 1-1correspondence between absolute values on K extending the usual absolutevalue on Q, modulo conjugation.

Proof. Let σ : K → C be an embedding. Define the absolute value | . |σ on Kby |x |σ = |σ(x) |, taking the usual absolute value of σ(x) ∈ C. This definesan archimedean absolute value on K extending the usual absolute value onQ. For surjectivity, suppose | . | is any absolute value on K extending theusual absolute value on Q. Let F be the the completion of K with respect to| . |. As F is either R or C we have an embedding σ : K → C and |x | = |σ(x)|.For injectivity, let σ1, σ2 be two embeddings such that the absolute values| . |σ1 , | . |σ2 are equivalent. Let Kσi be the completion of K with respectto | . |σi . For i = 1, 2, define the embeddings σi : Kσi → C by σi(x) =limn→∞

σi(xn) where {xn} is a sequence in K converging to x. Each σi fixesthe completion R of Q and we have a commutative diagram

K

σ1

��

σ2

&&MMMMMMMMMMM

C ⊇ Kσ1// Kσ2 ⊆ C

giving a continuous isomorphism Kσ1 7→ Kσ2 fixing R. Either both Kσ1 , Kσ2

are R or both are C. Hence σ1 and σ2 are equal to the identity or areconjugate of each other 1 . This shows injectivity since a complex embeddingσ and its conjugate σ induces the same absolute value.

1In fact there is no non-trivial automorphism of R at all, continuous or not. If we


The proposition 3.2.9 and theorem 3.1.26 give the complete list of abso-lute values on a number field K/Q.

We just saw that the complete archimedean fields arising from numberfields have a much simpler characterization. Now what about the non-archimedean complete fields?

Let K be a valued field with a non-archimedean absolute value | . |. Letv( . ) = − log(| . |) be the corresponding valuation. For our purposes, weshall be concerned with discrete valuations v : K∗ → eZ with e ∈ N andsurjective. As usual we call A the discrete valuation ring of v, m its onlymaximal ideal and k = A/m its residue field. Let π ∈ A with minimalpositive v-value, then it is easy to see that m =< π > . A generator π ofm is called a prime of K or uniformizer , it is characterized by v(π) = e.As A is a discrete valuation ring, one knows all the ideals p are of the formp = πnA = mn for some positive integer n. Hence we have the filtration ofideals of A

A ⊃ πA ⊃ π2A ⊃ · · · ⊃ πnA ⊃ · · · .

Note that the πnA are both open and closed in the v-topology as x ∈ πnA⇔v(x) ≥ nv(π)⇔ v(x) > (n− 1)v(π) and they form a fundamental system ofneighborhoods of zero.

In the multiplicative group K∗, inside the group of units A∗, we have thegroup of principal units U1 = 1+πA and the n-th higher unit group Un =1 + πnA. These are both closed and open and form a fundamental systemof neighborhoods of 1. We also have the filtration of subgroups

A∗ ⊃ 1 + πA ⊃ 1 + π2A ⊃ · · · ⊃ 1 + πnA ⊃ · · · .

The successive quotients in this filtration are computed as follows.

Proposition 3.2.10. With the above notation one has

A∗/Un ∼= (A/πnA)∗, Un/Un+1∼= A/πA ∼= πnA/πn+1A, for n ≥ 1.

Proof. The surjective ring homomorphism A 7→ A/πnA induces the homo-morphism of groups of units φ : A∗ → (A/πnA)∗, u ∈ A∗ 7→ φ(u) := u(mod πn). φ is surjective and has kernel Un, hence A∗/Un ∼= (A/πnA)∗.Next, define χ : Un → A/πA, u = 1 + aπn ∈ U1 7→ χ(u) := a (mod π).

suppose the automorphism to be continuous, then it is the identity on the dense set Q;thus it is the identity on R. But we can get rid of the continuity hypothesis by using thefact that R is totally ordered. Indeed, let ψ be an automorphism of R, then ψ preserves theorder on R. Indeed let x < y be real numbers. Then there is w ∈ R with y−x = w2. Thusψ(y)−ψ(x) = ψ(w)2. Now choose a ∈ R such that a < ψ(a) = b. Hence there is a rationalnumber r with a < r < b, this gives ψ(a) < ψ(r) = r < ψ(b), and this is absurd. For theautomorphisms of C, the identity and the complex conjugation are the only continuousautomorphisms while there are uncountably many discontinuous automorphisms, see [24]for the details.


This is a surjective homomorphism with kernel Un+1. Lastly, define ψ :A/πA → πnA/πn+1, a + πA 7→ πna + πn+1A. This is also onto and injec-tive.

Let now (K, v) be the completion of (K, v). In contrast with (R, | . |∞),(Q, | . |∞) where the value group |R∗ |∞ = R>0 ⊃ |Q∗ |∞ = Q>0, the valuegroup of K and its completion K are the same.

Lemma 3.2.11. Let (K, v) be the completion of (K, v), then v(K∗) =v(K∗).

Proof. It is clear that v(K∗) ⊂ v(K∗). Now let x ∈ K∗. Then there is a v-Cauchy sequence {xn} in K that converges to x. So there exists n0 ∈ N suchthat for all n ≥ n0 we have v(x−xn) > v(x). Then we have v(xn−x+x) =min{v(xn − x), v(x)} = v(x).

In other words, the value group of a discrete valued field K is invariantunder completion. The residue field is also invariant under completion.

Proposition 3.2.12. Let (K, v) be the completion of (K, v) with respectivevaluation rings A, A, maximal ideals m, m and residue fields k, k. Then

A/mn ∼= A/mn for n ≥ 1.

In particular k ∼= k.

Proof. Since mn ∩ A = mn we have an embedding φ : A/mn ↪→ A/mn. Forsurjectivity, let x ∈ A, then there is a Cauchy sequence {xm} in K suchthat x = lim

m→∞xm. We have v(x) ≥ 0, hence by continuity of v, v(xm) ≥ 0

for large m. So, for large m, xm ∈ A and xm ≡ x (mod mn). This givesφ(xm) = x.

We shall next see that our informal definition of the p-adic number fieldin the previous chapter is a special case of a general fact, that is to say thatthe elements in a complete non-archimedean field admit a representation asa convergent Laurent series in a prime element. Let R be a subset of Asuch that the restriction of the canonical projection A → A/m is bijectiveand 0 ∈ R. R is called a system of representatives of k = A/m in A. Bydefinition of R we have A =

⋃r∈R{r+ m} = {r+ m : r ∈ R}. Hence one has

A = R+ m. For n ∈ Z, let πn be such that v(πn) = n.

Theorem 3.2.13. Let (K, v) be the completion of (K, v) and let πn ∈ Kbe as above. Then any x ∈ K has a unique representation as a convergentLaurent series expansion:

x =∑n≥n0

rnπn, with n0 ∈ Z, rn ∈ R.


Proof. Existence: From the equality A = R+m, and by noting that πnA =mn, one has mn = πnR+ πn+1R+ · · ·+ πmR+ mm+1 for m ≥ n. Hence forany x ∈ mn, and for m ≥ n there exist rn, rn+1, · · · , rm ∈ R such that

x ≡ rnπn + · · ·+ πm (mod mm+1).

This gives x =∑∞

n≥n0πnrn.

Uniqueness: Suppose that we have 0 6= x =∑

n≥n0rnπn =

∑i≥i0 aiπi

with ai0 , rn0 6= 0 and ai ∈ R. Then i0 = v(x) = n0. From 0 = (ai0−ri0)πi0 +∑i≥i0+1(ai − ri)πi, one sees that we must have v(ai0 − ri0) = ∞, that is

ai0 = ri0 . Therefore one obtains that ai = ri for all i ≥ i0 and so one getsthe uniqueness of the expansion of x.

In particular, if we take K = Q, and πn = pn, we see that the comple-tion of Q with respect to the p-adic valuation is just Qp the field of p-adicnumbers.

This representation of complete discrete valued fields leads to an alter-native way of viewing them. Let π be a prime of K, then one can takeπn = πn. Therefore any a ∈ A can be written as a =

∑∞i=0 anπ

n. Con-sider now the partial sums sn =

∑n−1i=0 anπ

n, we have sn+1 ≡ sn (mod πn).Hence, (s1 (mod m), s2 (mod m2), · · · , sn (mod mn). · · ·) ∈ lim

←nA/mn ,

the projective limit. Thus we have an homomorphism

ψ : A → lim←n

A/mn

ψ(a =∞∑i=0

anπn) 7→ (s1 (mod m), · · · , sn (mod mn), · · ·).

Proposition 3.2.14. ψ is an isomorphism of topological rings. Here A isendowed with the topology induced by the valuation and the projective limitis endowed with the topology induced by the product topology of

∏nA/m

n

where the finite rings A/mn are endowed with the discrete topology. Takingunits yields the isomorphism:

(A)∗ ∼= (lim←n

A/mn)∗ ∼= lim←n

(A/mn)∗ ∼= lim←n

A∗/Un.

Proof. See [18, p 128].

This proposition 3.2.14 is a special case of the more general notion ofm-adic completion. See [2], [4] or [6] for the details on this notion. Oneimportant advantage of using projective limits is that arithmetic in lim

←iA/mi

is easier as it comes down to arithmetic modulo mi for all i ≥ 1.We shall now turn to the task of finding roots of polynomials over a

discrete valued field. We divert briefly to the similar problem in R becausethe notion that follows is similar to what we know for R. Let f(X) be a


polynomial of R[X]. We ask for a real root of f if it exists. As finding aroot is a hard exercise, one is often satisfied with an approximation to such aroot. One has at hand Newton’s method for this problem. Newton’s methodrelies on the sequence of real numbers {xn}n∈N where xn+1 = xn − f(xn)

f ′(xn) ,with f ′ the derivative of f and f ′(xn) 6= 0 for each n ∈ N. It easy to seethat in the case where the Newton sequence converges in R, then its limitis a root of f . Unfortunately, this sequence does not always converge in Rsee [12, p 166]On the other hand, when f(X) is a polynomial in Zp[X], we saw that froma simple root α of f modulo p, we can lift it to a root of f in Zp. We takeα as first approximation of the root and form the Newton sequence {xn}with x0 = α, it will necessarily converge in Zp as already seen in the proofof proposition 2.1.7. We shall give this method of lifting a root in a moregeneral context.

Proposition 3.2.15. (Hensel’s lemma: Root Form) Let (K, | .|) be acomplete discrete valued field with valuation ring A and maximal ideal m.Let f(X) =

∑mi=0 amX

m ∈ A[X] and suppose that there is α0 ∈ A such that| f(α0) | < |f ′(α0)2|. The following holds:

1. There exists an α ∈ A with f(α) = 0 and |α− α0| < 1.

2. Furthermore if |f ′(α0)| = 1, then such an α is unique.

Proof. 1. For n ≥ 0, consider Newton’s sequence αn+1 = αn − f(αn)f ′(αn) .

Introduce also, the sequence of positive real numbers un = | f(αn)f ′(αn) |.

Firstly for n ≥ 0, we have |f(αn)| < |f ′(αn)2|. Indeed, this is truefor n = 0, by assumption, suppose then that |f(αn)| < |f ′(αn)2|.By Taylor’s expansion theorem we have f(αn+1) = f(αn − f(αn)

f ′(αn)) =

f(αn) − f(αn)f ′(αn)f

′(αn) + f(αn)2

f ′(αn)2f ′′(αn)

2! + · · · + (−1)m f(αn)m

f ′(αn)mf (m)(αn)

m! =f(αn)hn where hn ∈ A. Hence |f(αn+1)| ≤ |f(αn)|. Also by Taylor’stheorem we have f ′(αn+1) = f ′(αn− f(αn)

f ′(αn)) = f ′(αn)− f(αn)f ′(αn)f

′′(αn)+

· · · + (−1)m−1 f(αn)m−1

f ′(αn)m−1f (m)(αn)(m−1)! = f ′(αn)( 1

f ′(αn) + h′n), with h′n ∈ A.

Thus we have |f ′(αn+1)| ≥ |f ′(αn)|. So, we obtain that | f(αn+1)f ′(αn+1)2

| ≤| f(αn)f ′(αn)2

| < 1. We deduce at the same time that the sequence αn iswell defined that is to say f ′(αn) 6= 0 for n ≥ 0 and also that thesequence un is a decreasing sequence which is bounded below by zeroand so un converges. Therefore, {αn} is a Cauchy sequence in thecomplete valuation ring A, so it converges say to α ∈ A and we havef(α) = 0. Next for n ≥ 1, write αn = α0−( f(α0)

f ′(α0) + · · ·+ f(αn)f ′(αn)). Hence

|αn − α0| ≤ | f(α0)f ′(α0) | < 1 for all n ≥ 1. Taking limit as n → ∞ we get

|α− α0| < 1


2. Now suppose that there is β ∈ A such that f(β) = 0 and |β − α0| < 11 . By Taylor’s theorem we have:0 = f(α0 + α− α0) = f(α0) + (α−α0)f ′(α0)

1! + · · ·+ (α−α0)mf (m)(α0)m!

0 = f(α0 + β − α0) = f(α0) + (β−α0)f ′(α0)1! + · · ·+ (β−α0)mf (m)(α0)

m! . Bythe formula an−bn = (a−b)(an−1 +an−2b+ · · ·+abn−2 +bn−1), notingthat α − α0, β − α0 ∈ m and after subtracting the second equationfrom the first, we obtain the equation

(α− β)(f ′(α0) + ω) = 0

with ω ∈ m. Since f ′(α0) is a unit, we deduce that we must haveα = β.

As a corollary we have

Corollary 3.2.16. Let f(X) ∈ A[X] and suppose that f has a simple rootλ modulo m. Then, f has a root α ∈ A with α = λ.

Proof. Let α0 be a lift of λ in A. As λ is a simple root of f we have f ′(α0) 6≡ 0(mod m). Hence |f ′(α0)| = 1 and |f(α0)| < |f ′(α0)2| = 1.

Remark 3.2.17. In fact proposition 3.2.15 and corollary 3.2.16 are equiva-lent. These are two of the various equivalent formulations of Hensel’s lemmato be found in the literature. One other formulation of Hensel’s lemmathat we shall see in the sequel is the uniqueness of an extension of a non-archimedean absolute value on K to any of its algebraic extensions, for moredetails see [14].

Keeping the same notation as in proposition 3.2.15, suppose in additionthat K has finite residue field with q = card(k). Then the polynomialXq −X has q − 1 non-zero simple roots in k and hence by Hensel’s lemmait has q − 1 distinct roots in A. Therefore A contains the (q − 1)-th rootsof unity and so A contains a primitive (q − 1)-th root of unity as any finitesubgroup of (K∗, .) is cyclic. In particular Zp contains a primitive (p−1)-throot of unity. Thus we have proved

Proposition 3.2.18. If the discrete valuation ring A has finite residue fieldwith q elements then A contains a primitive (q − 1)-th root of unity.

Keeping the same assumption as in the above proposition 3.2.18, let πbe a prime in A and let K be the field of fractions of A. Let ζ be a primitive(q−1)-th root of unity. Consider the set R :=< ζ > ∪{0}. Then R is a set of

1Once we know that the absolute value onK can be extended on any algebraic extensionof K, then we don’t need to impose β ∈ A. The same argument holds for β lying in analgebraic extension of K.


representatives for A/πA. It is closed under multiplication and each elementx ∈ K∗ can be written as x =

∑∞n≥n0

ωnπn with ωn ∈ R from the expansion

theorem. The set R is called the set of the Teichmuller representatives.

Chapter 4

Algebraic extensions ofcomplete valued fields

In this chapter, we start with the question of extending an absolute value toan algebraic extension of a complete valued field. Then, we will define andfocus on the main object of this thesis, local fields by giving an account oftheir Galois and ramification theory. The last section is concerned with thenorm group of a local field. The content for this chapter is inspired from[18], [1], [23] or [8].

4.1 Extending absolute values

Let K be a complete field with absolute value | . |K . Here we are concernedwith the problem of extending the absolute value | . |K of K to an algebraicextension E/K. As an infinite algebraic extension is the union of its finitealgebraic subextensions, we first restrict to finite extensions. So, we assumethat E/K is finite. The case where | . |K is archimedean, is handled as fol-lows. C is algebraically closed, so the algebraic extensions of R are C or R.So, by Ostrowsky’s theorem 3.2.8, p 25, we are done.We now suppose that the absolute value | . |K is non-archimedean and dis-crete. By the following remark we can assume the finite algebraic extensionE/K to be separable.

Remark 4.1.1. We have to worry about separability only in characteristicp when E/K is purely inseparable. Then [E : K] = q, a power of p, and∀x ∈ E, xq ∈ K. Hence the rule |x |E = (|xq |K)

1q defines the extension of

| . |K .

So, our finite algebraic extensions E/K are assumed to be separableunless otherwise stated. We assume in addition that K has finite residuefield. We first remark

33

34 Algebraic extensions of complete valued fields

Remark 4.1.2. Let A be the valuation ring of K and m its maximal ideal.Then, the quotients A/mn for n ≥ 2 are finite because of the finiteness ofthe residue field A/m. Indeed, let π ∈ A be a prime and let R be a set ofrepresentatives for A/m in A. Then for n ≥ 2, each α ∈ A can be uniquelywritten as α = a0 + a1π + · · · + an−1π

n−1 with ai ∈ R. Therefore we havea bijection A/mn → (A/m)n and so A/mn is finite with cardinal qn where qis the cardinal of A/m.

Then for the ground field K, we have

Proposition 4.1.3. Let K be a complete discrete valued field with finiteresidue field, valuation ring A and maximal ideal m. Then

(a). A is compact;

(b). K is locally compact.

Proof. The local compactness of K is a consequence of the compactness of Asince for each α ∈ K, α+A will be a compact neighborhood of α. From thehomeomorphism A ∼= lim

←nA/mn, we have that A is compact since lim

←nA/mn

is closed in the compact space∏∞n=1A/m

n ( product of finite rings, hencecompact spaces), so it is also compact.

We shall show that there exists an extension of | . |K to E and it isunique. Suppose that we have a unique extension | . |E of | . |K , and assumethat E/K is Galois. Then for σ ∈ Gal(E/K), | . |σ = | . |E and hence we

have the formula |x |E = |NE/K(x)|1nK with [E : K] = n. This means that

if a unique extension of | . |K to E exists, then it is defined by this formula.

We are left to show now that |x |E = |NE/K(x) |1nK is indeed the unique

extension of | . |K to E.

Theorem 4.1.4. With the above notations, then |x |E = |NE/K(x) |1nK is

the unique absolute value on E extending | . |K .

Proof. We first check that this is an absolute value. For x, y ∈ E, |xy |E =|x |E | y |E from the definition and |x |E = 0 ⇔ x = 0 similarly. Now,we have to verify that | . |E satisfies the ultrametric inequality |x + y |E ≤max{|x |E , | y |E} or equivalently that | γ + 1 |E ≤ 1 if | γ |E ≤ 1. So, letf(X) = Xd + ad−1X

d−1 + · · ·+ ao be the minimal polynomial of γ over K.Consider the extension K(γ)/K. From NE/K(x) = NK(γ)/K(NE/K(γ)(x)),one has NE/K(γ) = (NK(γ)/K(γ))[E:K(γ)]. Therefore |NE/K(γ) |K ≤ 1 ⇔|NK(γ)/K(γ)|K ≤ 1. So, it comes down to showing the inequality forK(γ)/K.To this end, the minimal polynomial of γ+1 is then g(X) = f(X−1) so thatNE/K(γ+1) = ±g(0) = ±((−1)d+ad−1(−1)d−1+· · ·+a1(−1)+a0). We needthe implication | a0 |K = |NK(γ)/K(γ) |K ≤ 1⇒ | (−1)d+ad−1(−1)d−1+· · ·+a1(−1) + a0 |K = |NK(γ)/K(γ + 1) |K ≤ 1 to hold. This is a consequence of

4.1 Extending absolute values 35

the lemma 4.1.5 below, a corollary of Hensel’s lemma.For the uniqueness part, let A be the valuation ring of K and let O be theintegral closure of A in E. Then O is the valuation ring for | . |E . Indeed bydefinition of the integral closure any a ∈ O satisfies NE/K(a) ∈ A. On theother hand if x ∈ E has minimal polynomial f(X) = Xd+ad−1X

d−1+· · ·+a0

over K, then NE/K(x) = ±a[E:K(x)]0 . So, if NE/K(x) ∈ A then so is a0. By

lemma 4.1.5 we obtain f(X) ∈ A[X]. This shows O = {x ∈ E : NE/K(x) ∈A} = {x ∈ E : |x |E ≤ 1}. Now suppose | . |′ is another absolute valueon E extending | . |K and let O′ be its valuation ring, we next show thatO ⊂ O′ as this implies that |.| = |.|′. Indeed the inclusion O ⊂ O′ isequivalent to the statement |x| ≤ 1 ⇔ |x|′ ≤ 1 and so their equivalence asabsolute values. Since they have the same restriction on K, one deducesthat |.| = |.|′. So, to see the inclusion O ⊂ O′ suppose that there existsx ∈ O r O′. As x is integral over A, there exist a0, · · · , an−1 ∈ A suchthat xn + an−1x

n−1 + · · · + a0 = 0. Also x 6∈ O′, so x−1 ∈ m′, the maximalideal of O′. Therefore, 1 + an−1x

−1 + · · · + a0(x−1)n = 0, which gives thecontradiction 1 ∈ m′. Thus, we must have O ⊂ O′.

Lemma 4.1.5. Let f(X) = anXn + an−1X

n−1 + · · · + a1X + a0, be anirreducible polynomial in K[X]. Then | an + an−1 + · · · + a1 + a0 |K =max{| an |K , | a0 |K}.

Proof. By contradiction suppose the statement is false and let ar be the firstfrom a1, · · · , an−1 with | ar |K = max{| a1 |K , · · · , |an−1|K}. Then a−1

r f(X) ∈A[X]. Let u ∈ A∗ be any unit and set g(X) = a−1

r f(X)−uX. Then g(X) ≡X(ana−1

r Xn−1 + · · ·−u) (mod m) so that 0 is simple root of g(X) modulom and hence by corollary 3.2.15, p 29, there are α ∈ m, h(X) ∈ A[X] suchthat g(X) = (X −α)h(X). Hence, a−1

r f(X) = (X −α)h(X) + uX. We thenobtain a−1

r f ′(X) = h(X)+(X−α)h′(X)+u. Note that the constant term ofh(X) is b0 := a−1

r a0α−1, so we have finitely many possibilities for b0. If b0 is

not a unit, then a−1r f ′(0) = b0 +(−α)h′(0)+u is unit. Hence by proposition

3.2.15, p 29, we see that a−1r f(X) is has a zero in A contradicting the fact

that f(X) is irreducible. Next if b0 is a unit we choose u such that b0 + uis a unit and we apply proposition 3.2.16 to deduce that f(X) would bereducible. So, in all cases we obtain a contradiction and this completes theproof of the lemma.

Remark 4.1.6. From the definition of | . |E , we see that if | . |K has asso-ciated valuation v, then v has as unique extension w given by the formulaw(x) = 1

[E:K]v(NE/K(x)), for each x ∈ E, and w is also discrete.

Continuing this discussion we shall next see that the field E is completewith respect to | . |E . In fact, we shall prove the completeness of E from localcompactness of E. In the archimedean case we know already that the finitealgebraic extensions are complete, so we assume that the absolute values are


discrete as usual. Let B be the discrete valuation ring of E which is also theintegral closure of A in E as saw in the proof of theorem 4.1.4, p 34. Sincethe extension E/K is finite of degree n = [E : K], B is a free A-module ofrank n, see [18, p 12] for the details.

Proposition 4.1.7. Let E/K be a finite extension of discrete valued fieldswith | . |E the unique absolute value on E extending | . |K and K complete. LetB,A be the valuation rings of E,K respectively. Then E is locally compactwhich implies that it is complete.

Proof. Local compactness: Similarly as in proposition 4.1.3, local com-pactness of E follows from the compactness of B. To see that B is compact,fix a basis β1, · · · , βn of B over A. Define the map:

ψ : An → B

ψ((α1, · · · , αn)) = α1β1 + · · ·+ αnβn

An endowed with the norm | (α1, · · · , αn) | := max{|α1 |K , · · · , |αn|K } be-comes a topological space. It’s in fact the product topology on An. Next Bis endowed with the topology induced by the absolute value of B extendingthat of A. Then, ψ is continuous and bijective. Hence B is compact as theimage of the compact space An under ψ. Therefore, E is locally compact.Completeness : Recall that for r ∈ R>0, the sets Vr = {x ∈ E : |x |E < r}form a fundamental system of neighborhoods of zero in E. Let {xn} bea Cauchy sequence in E. Then we know that it is bounded, that is thereexists M ∈ R>0 such that |xn | ≤ M for all n ∈ N. Consider the subsetS = {x ∈ E : |x |E ≤ M}. By the local compactness of E, there existsCr = {x ∈ E : |x |E ≤ r}, a compact neighborhood of zero.Therefore bychoosing α ∈ E∗ with |α |E ≥ r−1M, we see that S is contained in the com-pact set αCr. Thus {xn} is a Cauchy sequence inside a compact subspace,so xn converges.

Remark 4.1.8. Commonly, the completeness of a finite extension E/Kof a complete field K is deduced from a general result on normed vectorspaces over complete fields: Let V be a finite dimensional vector space overa complete field K; it is a theorem that all the norms on V are equivalent tothe sup-norm, see [18, p 132] for the details.

We now turn to a converse of proposition 4.1.3. Let us first define themain object of study in this thesis.

Definition 4.1.9. A local field is a complete discrete valued field with finiteresidue field.

Example 4.1.10. 1. The p-adic field Qp is the prototype of a local fieldof characteristic zero.

4.1 Extending absolute values 37

2. More general than Qp, the completion of a number field K at a primeideal p, Kp, is a local field.

3. The field of formal Laurent series Fq((X)) in the variable X over afinite field Fq, is a local field of positive characteristic.

Soon, we will see that the local fields are exactly the finite extensions ofQp or Fq((X)). To this end, we will need the following result, which gives atopological criterion for a field to be local.

Theorem 4.1.11. Let K be a field with a valuation v, not necessarily dis-crete. Then K is locally compact if and only if the following hold:

1. K is complete;

2. the valuation ring A is compact and the residue field is finite;

3. v is a discrete valuation.

Proof. We have proved in proposition 4.1.3 that the enumerated propertiesimply that K is locally compact.For the converse let K be locally compact. We have showed in proposition4.1.7 that K is complete. By local compactness, there exists W a compactneighborhood of zero. Hence, there exists α ∈ K such that αA ⊂ W. SinceA is closed, αA is closed in the compact subset W, so it is compact. Wededuce that A is compact. As each ideal a of A is at the same time openand closed, the quotient topology on the ring A/a is the discrete topology.This, with the compactness of A implies that A/a is compact and discrete,hence finite. In particular, the residue field A/m is a finite field. For α ∈ A,the ring A/αA is finite, this means that there are only a finite number ofideals of A that contain α. So the set Pv(α) = {v(x) : x ∈ A, 0 < v(x) <v(α)} must be finite. Otherwise, we would have infinitely many xA withA % xA % αA. Therefore, Pv(α) has a least element, say λ. Choose π ∈ Awith v(π) = λ. For each x ∈ A with v(x) > 0, we obtain v(π) ≤ v(x),so that v(π) = λ is the least element in the set {v(x) > 0 : x ∈ A}.Now, let m ∈ N be the greatest integer such that mv(π) ∈ Pv(α). Thenmλ < v(α) < (m+ 1)λ, which gives 0 < v(α)−mλ < λ. Therefore, by thedefinition of λ, we must have v(α) = mλ. And hence v(K∗) = λZ.

From now on, unless otherwise stated, all fields are local. Next we definesome invariants for an extension of local fields E/K. From the inclusionE∗ ⊃ K∗, we get that vK(K∗) is a subgroup of vE(E∗) where vE is the uniquevaluation extending vK . As vE(E∗) is discrete, the index (vE(E∗) : vK(K∗))is finite as nontrivial subgroups of Z have finite index. Define e(E/K) to bethe index (vE(E∗) : vK(K∗)). Also, the residue fields are finite so [kE : kK ]is finite. Set f(E/K) = [kE : kK ]. So e(E/K) and f(E/K) are finite forany extension E/K of local fields.


Definition 4.1.12. e(E/K) is called the ramification index of the extensionE/K. f(E/K) is the residue degree of the extension E/K.

Remark 4.1.13. If we fix prime elements πK , πE of K,E respectively, thene = e(E/K) is the integer such that πKB = πeEB.

Lemma 4.1.14. For an intermediate field K ⊆ F ⊆ E one has

f(E/K) = f(E/F )f(F/K), e(E/K) = e(E/F )e(F/K).

Proof. These relations follows from the definition.

Given an extension E/K of local fields, let A,B, be the valuation ringsof K,E respectively. The expansion theorem 3.2.13, p 27, allows one to findan explicit integral basis for B over A. Then it follows that any extensionE/K of local fields is finite since a basis for B over A is also a basis for Eover K.

Proposition 4.1.15. Let E/K be an extension of local fields. We writef = f(E/K), e = e(E/K). Take u1, · · · , uf ∈ B such that their reductionsmodulo mE is a basis for kE over kK . Fix a prime element πE ∈ B. Then,the elements πjEui, 0 ≤ j ≤ e − 1, 1 ≤ i ≤ f, form an integral basis for Bover A, so that ef = [E : K].

Proof. For n ≥ 0, write n = qe + s with 0 ≤ s < e and set πn = πqKπsE .

Then vE(πn) = qe + s = n. Let also RA be a set of representatives for kKin A. The set RB = RAu1 + · · · + RAuf is a set of representatives for kEin B. Indeed, card(RB) = (card(kK))f and each element in B is congruentmodulo mE to some β ∈ RB. Therefore by the expansion theorem, for β ∈ Bwe have

β =∞∑q=0

e−1∑s=0

πqKπsE(

f∑i=0

rq,r,iui), with rq,s,i ∈ RA

=f∑i=1

e−1∑s=0

πsEui

∞∑q=0

πqKrq,s,i.

Each∑∞

q=0 πqKrq,s,i ∈ A, so B is finitely generated as an A-module. Hence

E/K is a finite extension. But A is a principal ideal domain and B is theintegral closure of A in E. So we know that B is a free A-module of rankn = [E : K]. Also B/πKB ∼= B/πeEB so that card(B/πKB) = (card(kK))ef .Therefore, one has ef = n. It follows that πjEui with j = 0, · · · , e − 1, i =1, · · · , f, form an integral basis for B over A. This completes the proof ofthe proposition.

We can now give the characterization theorem for local fields.

4.2 Galois theory and the norm group of local fields. 39

Theorem 4.1.16. A local field is a finite extension either of Qp or of thefield of formal Laurent series Fp((X)).

Proof. It is clear that Qp and Fp((X)) are local fields. From theorem 4.1.11any finite extension K of one of the local fields Qp or Fq((X)) is a local field.

Conversely, let K be a local field. If K is of characteristic zero, thenK contains the prime field Q. Let p be the characteristic of the residuefield A/mK , where A is the valuation ring of K, and mK its maximal ideal,then we have p ∈ mA. Therefore the restriction of the valuation v of Kto Q is equivalent to the p-adic valuation vp, hence the closure of Q in Kis Qp. Thus we have an extension K/Qp of local fields. From proposition4.1.15, K/Qp is finite 1 . Now, if K is of positive characteristic p, thenp is also the characteristic of its residue field kK = Fq ( p = 0 in K ⇒p = 0 in kK ), with q = pf , f = [Fq : Fp]. Then K ∼= Fq((X)) ⊃ Fp((X)).Indeed, let R = {ζnq−1 : n ∈ Z}

⋃{0} = {roots of Xq −X} be the set of the

Teichmuller representatives of kK = Fq in A, see p 30. But also we haveFq ∼= {roots of Xq−X} in A. Therefore, R ∼= Fq. Fixing a prime π ∈ K andusing the expansion theorem, we see that K is isomorphic to Fq((X)) withπ ↔ X. So, K is a finite extension of Fp((X)).

We shall now turn to the study of Galois extensions of local fields.

4.2 Galois theory and the norm group of localfields.

In this section we gather the basic facts concerning ramification theory forlocal fields. We introduce the higher ramification groups for a Galois exten-sion of local fields and finish with the basic facts about the norm group of alocal field.

4.2.1 Ramification in an extension of a local field.

Let E/K be a finite extension of local fields.

Definition 4.2.1. 1. If e = e(E/K) = 1, the extension E/K is said tobe unramified. We then have f = [kE : kK ] = [E : K] and πK =πEu with u ∈ A∗E .

2. If e = [E : K] so that f = [kE : kK ] = 1, we say that the extensionE/K is totally ramified.

3. If the characteristic of the residue field is p > 0 and does not dividethe ramification index e, then, the extension E/K is said to be tamely

1 The fact that [K : Qp] is finite is a general property of locally compact topologicalvector spaces, theorem 3 in [3, p TVS I.15].


ramified. We see immediately that an unramified extension is tamelyramified and a totally ramified extension may not be tamely ramified.

4. An extension which is not tamely ramified is said to be wildly ramified.

Remark 4.2.2. An extension E/K of local fields is unramified if and onlyif a prime πK ∈ K is also prime in E.

For an extension E/K of local fields, let πE , πK be primes of E, K re-spectively. From πK = πeEu where u ∈ A∗E , we obtain π

[E:K]K = NE/K(πK) =

(NE/K(πE))eNE/K(u). Therefore after normalizing vK , we have ef = evK(NE/K(πE)). From f = vK(NE/K(πE)) we deduce that E/K is totally ram-ified if and only if the norm of a prime in E is prime in K.

An infinite algebraic extension E/K of valued fields is in one of theclasses 1-4 above if all its finite subextensions are.

Proposition 4.2.3. Let E/K be a finite extension of local fields. ThenAE = AK [α] with some α ∈ AE .

Proof. Since kE/kK is separable and finite, kE = kK [β], for some β ∈ kE . Letf(X) ∈ kK [X] be the minimal polynomial of β over kK . Let g(X) ∈ AK [X]be monic such that its reduction modulo mK is f(X). As β is a simple root ofg(X) modulo mK , by Hensel’s lemma there exists α ∈ AE with α = β in kEand g(α) = 0. Let α1 = α + aπE with a ∈ AE and πE a prime of AE . Wehave g(α1) ≡ g(α) + aπEg

′(α) (mod π2E). As g′(α) 6= 0 modulo mE , we see

that we cannot have both g(α1), g(α) ≡ 0 (mod π2E). Hence by renaming

if necessary, we can assume that g(α) is a prime element of AE . Hence fromproposition 4.1.15, αig(α)j with 0 ≤ i ≤ f − 1, 0 ≤ j ≤ e − 1, is an AKbasis for AE .

Unramified extensions:

Let (K, v) be a complete discrete valued field with kK as residue field. Thenwe have

Theorem 4.2.4. 1. For each positive integer n there exists a field exten-sion F/K with [F : K] = n = f(F/K), e(F/K) = 1.

2. Such an extension F is unique inside an algebraic closure K of K.

Proof. Existence: Let F be the splitting field of Xqn−1−1 over K. Let ζ bea primitive (qn−1)-th root unity and let f(X) be the minimal polynomial ofζ over K. f(X) is a factor of Xqn−1 − 1 and f(X) is irreducible over kK [X]since otherwise Hensel’s lemma will show that f(X) is not irreducible. Weobtain [F : K] = deg(f) = deg(f) ≤ [kF : kK ] ≤ [F : K], so F/K isunramified.


Uniqueness: Let n = [E : K]. e(E/K) = 1 so that [kE : kK ] = n. WritekK = Fq, kE = Fqn . Fqn is the splitting field of Xqn−1 − 1. By Hensel’slemma, there exists ζ ∈ E a primitive (qn − 1)-th root of unity. Then K ⊂K(ζ) ⊂ E. Hence kK(ζ) = kE . Therefore [K(ζ) : K] ≥ [kE : kK ] = [E : K]and thus E = K(ζ).

Corollary 4.2.5. Let E/K be a finite extension of local fields. Then thereexists a unique subextension L/K such that L/K is unramified and E/L istotally ramified.

Proof. Let f = f(E/K). We take L as the unique unramified extension ofK of degree f.

Definition 4.2.6. This maximal unramified extension L/K is called theinertia field of the finite extension E/K.

Corollary 4.2.7. Let E′/K and L/K be algebraic extensions of a local fieldK. Let E = E′L. Then if L/K is unramified so is E/E′. The composite ofunramified extensions of a local field K is also unramified. Furthermore, thesubextensions of an unramified E/K are unramified.

Proof. Without loss of generality, we can assume all extensions finite. Wehave to check that [E : E′] = [kE : kE′ ]. As the extension L/K is unramified,we know that L = K(α) with α such that its reduction generates kL. Wethus obtain E = E′K(α) = E′(α). So, let h(X) ∈ AE′ [X] be the minimalpolynomial of α over AE′ . Let l(X) ∈ kE′ [X] be the minimal polynomial ofthe reduction of α over kE′ . We have deg(h) = deg(l). Indeed, any root ofh(X) reduces to a root of l(X) and since l(X) is separable, it follows thattwo distinct roots of h(X) reduces to two distinct roots of l(X). Thereforewe have

[kE : kE′ ] ≤ [E : E′] = deg(h) = deg(l) ≤ [kE : kE′(α)] ≤ [kE : kE′ ].

This means [E : E′] = [kE : kE′ ] and hence E/E′ is unramified.Let now L/K, L′/K, be unramified and put T = LL′. Then T/L′ is unram-ified, and from e(T/K) = e(T/L′)e(L′/K), we deduce that T/K is unram-ified. Next let L/K be an subextension of an unramified extension E/K.From 1 = e(E/K) = e(E/L)e(L/K), we see that e(L/K) = 1, i.e., L/K isunramified.

From the above we see that the union of all the unramified extensions ofa local field K inside an algebraic closure K of a K is a field. It is denotedby Kur and by construction it is the maximal unramified extension of K. Itis a natural object to consider as it allows one to handle the finite unramifiedextensions of K at once. From its definition we see that Kur is not a localfield as it is of infinite degree over K. Furthermore it fails to be complete


with respect to the unique discrete valuation extending vK , the valuation ofK. Nonetheless, the Hensel’s lemma does hold. This is an example of whatis called a Henselian field. For an introduction to the theory of Henselianfields, consult [18], [10], or [9].

Totally and wildly ramified extensions:

Let E/K be a totally ramified extension of local fields. We know fromproposition 4.1.15, that AE = AK [πE ] with πE a prime element of AE .Now [E : K] = e = e(E/K). The minimal polynomial of πE over K is sayP (X) = Xe + ae−1X

e−1 + · · ·+ a0 with ai ∈ AK . Since the extension E/Kis totally ramified, vK(NE/K(πE)) = vK(±ao) = f(E/K) = 1, that is a0 isa prime in AK . From lemma 4.1.5, p 35 we deduce that all ai ∈ mK . Sucha polynomial is known as an Eisenstein polynomial.Conversely let π be a root of an Eisenstein polynomial P (X) = Xn +an−1X

n−1 + · · · + a0 over AK with n ≥ 2. Let vK be the valuation onK. Consider the extension K(π) and let w be the unique extension of vK toK(π). Then from the integral relation

πn + an−1πn−1 + · · ·+ a1π + a0 = 0,

it follows that two of the terms must have the same minimal w value. As allvK(ai) > 0, we have w(π) > 0 and w(aiπi) > w(a0) for all i = 1, · · · , n− 1.This is because P (X) is Eisenstein. Therefore we deduce w(πn) = w(a0)which is equivalent to nw(π) = e. But we know that e ≤ n and hence wehave w(π) = 1, e = n. This means that K(π)/K is totally ramified and thatπ is a prime in K(π). We have proved.

Proposition 4.2.8. A finite algebraic extension E/K of local fields is totallyramified if and only if E = K(π) for a prime π in AE which is a root of anEisenstein polynomial in AK [X].

In the case of Qp, a totally ramified extension is constructed as follows.

Example 4.2.9. Let q = pn with p a prime, and ζq be a primitive q-th rootof unity. Consider the extension K = Qp(ζq). Write Xpn − 1 = (Xpn−1 −1)(X(p−1)pn−1

+ · · ·+Xpn−1+ 1). Then ζq is a root of h(X) = X(p−1)pn−1

+· · ·+Xpn−1

+ 1. By the change of variable X ↔ X+ 1, we see that h(X+ 1)is Eisenstein over Zp. Therefore K/Qp is totally ramified with degree (p −1)pn−1 = φ(pn). We also have AK = Zp[ζq]. Indeed λ = ζq − 1 is a rootof the Eisenstein polynomial h(X + 1), so it is a prime. Therefore we haveAK = Zp[λ] = Zp[ζq].

Next we shall see that a totally ramified extension E/K of local fieldscontains a maximal tamely ramified subextension. So a ramified extensionmay be split into a tamely and a wildly ramified extension.


Proposition 4.2.10. Let E/K be a totally ramified extension of local fieldswith p the characteristic of the residue field kK . Let n = [E : K] =n0p

k with (n0, p) = 1. Then there exists a unique subextension T/K insideE given by T = K( n0

√π) for some prime π ∈ K and [T : K] = n0.

Proof. Existence: As E/K is totally ramified, we have πnE = uπK with u ∈A∗E and πE , πK prime elements from AE , AK respectively. Since kE = kK ,there exists v ∈ A∗K such that u ≡ v (mod mE). Let α := v πK

(πpk

E )n0. The

polynomial f(X) = Xn0 − α has 1 as simple root modulo mE , whence byHensel’s lemma there exists a unique β ∈ AE such that f(β) = 0 andβ ≡ 1 (mod mE). Hence β(πE)p

kis a root of the Eisenstein polynomial

Xn0 − vπK ∈ AK [X]. That is the extension T = K(βπpk

E ) = K( n0√vπK)/K

is totally and tamely ramified extension of degree n0.Uniqueness: Suppose that we have T1, T2 two tamely ramified extensionsof K inside E of degree n0. By the above, we see that there exist π1, π2

primes of T1 and T2 respectively and πn01 = πK , π

n02 = vπK for some prime

πK ∈ AK and v ∈ A∗K . Then x = π1π2∈ A∗E and xn0 ∈ AK . From kE =

kK , there exists w ∈ A∗K such that x ≡ w (mod mE) ⇔ xw−1 ∈ 1 +mE , i.e, |xw−1−1|E < 1. We first observe that xw−1 is a root the polynomialf(X) := Xn0 − xn0w−n0 ∈ AK [X] with |xw−1 − 1|E < 1, but also thepolynomial f(X) has 1 as simple root modulo mK and hence by corollary3.2.16, it has a root β ∈ AK with |β − 1|K < 1. Since f ′(1) is a unit,proposition 3.2.15 says that xw−1 = β ∈ 1 + mK . This means that x = π1

π2∈

A∗K and hence T1 = T2.

Corollary 4.2.11. Let T/K, T ′/K be two extensions of K. Let E = TT ′.Then one has: T/K is tamely ramified ⇒ E/T ′ is a tamely ramified exten-sion. The composite of tamely ramified extensions is also tamely ramifiedand any subextension of a tamely ramified extension is tamely ramified.

Proof. We can assume the extensions to be finite. To simplify matters, weassume that T ′/K is Galois and [T : K] is prime to p 2 . It is clear that anysubextension of a tamely ramified extension is also tamely ramified. NowT/K is tamely ramified, since [E : T ′] = [T : T ∩ T ′], then E/T ′ is alsotamely ramified. Next, let T1, T2 be tame ramified extensions of K. ThenT1T2/T2 is tamely by the preceding argument and so from e(T1T2/K) =e(T1T2/T2)e(T2/K), we deduce that T1T2/K is also tamely ramified.

As a consequence, we see that the union inside a fixed algebraic closureof all finite tamely ramified extensions of a local field K is a field. Thisfield is denoted by Ktame and is of infinite degree over Kur. Ktame is calledthe maximal tamely ramified extension of K. This is also a Henselian field.Having this, we next give the Galois theory of extensions of local fields.

2This proposition is also true without these assumptions, the reader may consult [18,pp 155-157] or [10, p 45] for a precise description of a tame extension of a local field.


4.2.2 Galois theoretical aspects for local fields.

Let K ⊂ L ⊂ T ⊂ E be the tower of local fields with L, T being respectivelythe maximal unramified and the maximal tamely ramified extension of Kinside E. When the extension E/K is Galois, we shall give the Galois groupscorresponding to these subfields. After this we will give a brief account ofthe higher ramification subgroups inside Gal(E/K).

Let E/K be Galois with Galois group Gal(E/K). We know that conju-gate elements have the same valuation, hence we have σ(AE) = AE , σ(mE) =mE . This means that any σ ∈ Gal(E/K) induces an automorphism σ ∈Gal(kE/kK) by σ(α) := σ(α). Hence we can define

ψ : Gal(E/K) → Gal(kE/kK)ψ(σ) = σ.

This is a group homomorphism.

Proposition 4.2.12. Let E/K an unramified extension of local fields. ThenE/K is cyclic Galois. Furthermore the homomorphism ψ is an isomorphism.

Proof. By construction of the extension E/K, see theorem 4.2.4, E/K iscyclic Galois. We are left to prove that ψ is an isomorphism. We havekK = Fq and kE = Fqn with n = f(E/K) = [E : K]. Also there is a primitive(qn − 1)-th root of unity ζ ∈ E with E = K(ζ). Furthermore, the reductionζ ∈ kE of ζ modulo mE is such that ζ is a primitive (qn−1)-th root of unityin kE . So kE = kK(ζ). Now for σ ∈ Gal(E/K), ψ(σ)(ζ) = ζ ⇔ σ(ζ) ≡ ζ(mod mE) ⇔ σ(ζ) = ζ. Therefore ψ is injective. Since both Galois groupshave the same order f = [kE : kK ] = [E : K], ψ is an isomorphism.

Corollary 4.2.13. Let E/K be a finite Galois extension. Then ψ is surjec-tive and the maximal unramified extension L/K inside E is the fixed fieldof ker(ψ).

Proof. We can write ψ as the composite Gal(E/K)→ Gal(L/K)→ Gal(kE/kK) = Gal(kL/kK), where the first arrow is the restriction map. We obtain

Gal(E/K)/ker(ψ) ∼= Gal(L/K).

Thus by Galois theory we have Gal(E/L) = Ker(ψ), i.e., L = Eker(ψ).

The subgroup of Gal(E/K) corresponding to the inertia field is calledthe inertia subgroup of Gal(E/K) and it is denoted by IE/K := kerψ. SincekK = Fq, kE = Fqf where f = [kE : kK ], Gal(kE/kK) is a cyclic groupgenerated by the Frobenius kK-automorphism

ϕ : kE → kE , ϕ(α) = (α)q.


By the isomorphism Gal(L/K) ∼= Gal(kE/kK), there exists a unique K-automorphism ϕ of L that corresponds to ϕ. It generates Gal(L/K) and ischaracterized by

ϕ(a) ≡ aq (mod mL),∀a ∈ AL.

Definition 4.2.14. This automorphism ϕ ∈ Gal(L/K) of an unramifiedextension L/K is called the Frobenius automorphism of L/K.

For a finite Galois extension of local fields E/K, we wish to completethe Galois correspondence

E oo // 1

��T

OO

Gal(E/T ) =?//oo

��L

OO

Gal(E/L) = ker[Gal(E/K)→ Gal(kE/kK)]//oo

��K

OO

Gal(E/K)//oo

where respectively T, L are the maximal tamely ramified and maximal un-ramified extensions of K inside E. Let then e = e(E/K) = n0p

k = [E : L]with (n0, p) = 1 with n0 = [T : L] and pk = [E : T ]. By Galois theory we seethat the subgroup Gal(E/T ) is of order pk inside IE/K . Hence it is a p-Sylowsubgroup of IE/K . If there were more than one p-Sylow subgroups of IE/K ,by Galois correspondence we would have a field T1 6= T with [T1 : L] = n0

contradicting the uniqueness of the maximal tamely ramified subextensionof E/K. Therefore, there is a unique p-Sylow subgroup of IE/K , and it is thesubgroup of Gal(E/K) that fixes the maximal tamely ramified extension.From the maximality of the extension T/K, inside E, one sees that it mustbe Galois. We would like to have a more precise description of Gal(E/T ),so let us carry on this discussion. Put S = Gal(E/T ) ⊂ IE/K .

For σ ∈ IE/K , we obtain σ(π)π ∈ A∗E for a prime π ∈ AE . For another π1 =

uπ with u ∈ A∗E , we have that σ(π1)π1

= σ(u)σ(π)uπ , and also σ(u)

u ∈ 1+mE = U1.

Whence σ(π1)π1≡ σ(π)

π (mod U1). That is the map

θ : IE/K → A∗E/U1,

σ 7→ σ(π)π

(mod U1)

is independent of π. On the other hand θ(στ) = σ(τ(π))π = σ(τ(π))τ(π)

τ(π)π ≡


θ(σ)θ(τ) (mod U1), i.e., θ is a homomorphism of groups and its kernel is

ker(θ) = {σ ∈ IE/K : σ(π)− π ∈ m2E}.

We can thus identify IE/K/ker(θ) with a subgroup of A∗E/U1∼= k∗E . This

implies that the index (IE/K : ker(θ)) is coprime with p the characteristicof kE and so ker(θ) contains the unique p-Sylow S of IE/K .Conversely, since T/L is tamely ramified of degree n0, there exists a primeπT ∈ T such that πn0

T = πL for some prime of L. Then σ(πT )πT

is a n0-throot of unity for σ ∈ IE/K . On the other hand the extension E/T is totally

ramified of degree pk so that πT = vπpk

E for some πE of E prime and a unitv of AE . This means σ(πT )

πT≡ (σ(πE)

πE)pk

(mod mE) for σ ∈ IE/K . We thenhave

σ ∈ ker(θ)⇒ (σ(πE)πE

)pk ≡ 1 (mod mE)⇔ σ(πT )

πT≡ 1 (mod mE).

But σ(πT )πT

is a n0-th root of unity so we must have σ(πT ) = πT . Indeed,

we have (σ(πT )πT

)n0 = 1 and if σ(πT )πT

= 1 + aπrE with a ∈ A∗E then 1 =(1 + aπrE)n0 ≡ 1 + n0aπ

rE (mod πr+1

E ). But also (n0, p) = 1, so we musthave a = 0. This gives

S = ker(θ) = {σ ∈ IE/K : σ(πE)− πE ∈ m2E}.

More generally, one observes first that for σ ∈ G = Gal(E/K) we haveσ(AE) = AE , σ(mi

E) = miE for i ≥ 1. So, there is an induced map σ :

AE/miE → AE/m

iE with σ(α) = σ(α) where α = α + mi

E . Hence we candefine group homomorphisms

ψi : G → Aut(AE/mi+1E )

σ 7→ σ

For i ≥ 0, define Gi := ker(ψi) so that IE/K = G0 and S = G1.

Definition 4.2.15. The subgroups Gi, i ≥ 0, are called the higher rami-fication subgroups in lower numbering associated with the Galois extensionE/K.

These higher ramifications subgroups form a decreasing filtration on G :

1 = · · · = Gl ⊂ Gl−1 ⊂ · · · ⊂ G1 ⊂ G0 ⊂ G−1 = G.

By Galois correspondence they give rise to the tower of fields:

E ⊃ · · · ⊃ Ti ⊃ · · · ⊃ T ⊃ L ⊃ K.


Ti is called the i-th ramification subfield for the extension E/K. From propo-sition 4.2.3, p 40, we have AE = AK [α] for some α ∈ AE . Then these higherramification subgroups can be described in terms of α. We have

Lemma 4.2.16. Take α ∈ AE such that AE = AK [α] and σ ∈ Gal(E/K).Then vE(σ(α)− α) = Infa∈AEvE(σ(a)− a).

Proof. Let a ∈ AE , we have a = an−1αn−1 + · · · + a0, with ai ∈ AK . We

obtain σ(a)−a = an−1(σ(α)n−1−αn−1)+· · ·+a1(σ(α)−α). Since σ(α)k−αkis divisible by σ(α) − α for k ≥ 1, we obtain that σ(a) − a is a multiple ofσ(α)− α, and hence the relation follows.

One then defines iE/K(σ) = vE(σ(α) − α) for α a generator of the val-uation ring AE over AK . From lemma 4.2.16, we see that iE/K(σ) is inde-pendent of the choice of α. We have Gi = {σ ∈ G : iE/K(σ) ≥ i+ 1}. As theextension E/L is totally ramified, one has AE = AL[π] with π a prime ofAE , so one can compute the successive quotients in the filtration as follows.The computation establishes that a finite Galois extension of local fields issolvable. It provides also a short argument to see that G1 is the uniquep-Sylow of the inertia subgroup IE/K . We observe as well that in contrastwith the number fields where the roots of a polynomial of degree greater orequal to 5 cannot be given by radicals in general, roots of a polynomial overlocal fields can be expressed with radicals.

So, let E/K be a totally ramified Galois extension of local fields withAE = AK [π], where π a prime of AE . We have Gi = {σ ∈ Gal(E/K) :σ(π) − π ∈ mi+1

E }. Therefore for σ ∈ Gi, we obtain σ(π)π ∈ 1 + mi

E = Ui. Ifπ1 = uπ with u ∈ A∗E = U, the group of units of AE , we get σ(π1)

π1= σ(u)

uσ(π)π .

Thus σ(π1)π1≡ σ(π)

π (mod Ui+1). Therefore we can define maps that areindependent of π :

φi : Gi → Ui/Ui+1

σ 7→ σ(π)π

(mod Ui+1).

Recall from proposition 3.2.10, p 26, the isomorphisms U/U1∼= k∗E , Ui/Ui+1

∼=kE for i ≥ 1. Then one has

Theorem 4.2.17. 1. φi is a homomorphism with kernel Gi+1.

2. G0/G1 is cyclic of order prime with p and Gi/Gi+1, i ≥ 1 are elemen-tary abelian p-groups.

3. G = Gal(E/K) is a solvable group.

Proof. This follows from the above discussion, for more details see [20, p 65].


Remark 4.2.18. The second statement of theorem 4.2.17 clearly gives an-other argument to see that G1 is the unique p-Sylow of IE/K .

The ramification subgroups carry non-trivial information concerning thearithmetic of Galois extensions of local fields E/K. So, it is natural toask how the ramification groups behave under the operation of restrict-ing to a subgroup H of G = Gal(E/K) or the operation of taking quo-tients G/H when H is normal in G. In other words given a tower of fieldsE ⊃ L ⊃ K with L/K Galois, what are the relations between Gi, Hi =Gal(E/L)i and Gal(L/K)i = (G/H)i. For subgroups this is clear:

Proposition 4.2.19. Hi = Gi ∩H.

Proof. This follows from the definition.

On the other hand when taking quotients things are not so simple. By thecanonical projection Gi maps to GiH/H, but in general (G/H)i 6= GiH/H.Thus, the numbering changes by passage to quotient. Writing GiH/H,and (G/H)j , for respectively the image of Gi modulo H and a ramificationsubgroup of G/H, we seek a relation between i and j. To this end a slightmodification of the definition of the ramification groups is needed. Theindexing set is made to be continuous as follows. For a real number t ∈[−1,+∞), one defines

Gt = {σ ∈ G = Gal(E/K) : iE/K(σ) ≥ t+ 1}.

Then we have G−1 = G, Gt = G0 for − 1 < t ≤ 0, Gt = G1 for, 0 < t ≤1 and in general Gt = Gn for n− 1 < t ≤ n. Given a normal subgroup H ofG, let L = EH . For τ ∈ G, and σ ∈ G/H = Gal(L/K), we write τ → σ tomean that the restriction of τ to L gives σ. Then we have

Proposition 4.2.20. Let H be a normal subgroup of G, and L = EH . Thenfor σ ∈ Gal(L/K), one has

iL/K(σ) =1

e(E/L)

∑τ→σ

iE/K(τ).

Proof. Write AE = AK [α], AL = AK [β] for α ∈ AE , β ∈ AL. By definitionfor σ ∈ Gal(L/K), we have iL/K(σ) = vL(σ(β)− β) and for τ ∈ Gal(E/K)we have iE/K(τ) = vE(τ(α) − α). Suppose τ → σ, the other elements inGal(E/K) which restrict to σ are of the form τh with h ∈ H. As for y ∈ Lwe have vE(y) = e(E/L)vL(y), we see that we need to check that

vE(σ(β)− β) = vE(∏h∈H

(τh(α)− α)).

If σ = 1, both sides have valuation equal to ∞. So, suppose σ 6= 1 and puta = τ(β)− β, and b =

∏h∈H(τh(α)− α). Next observe that a, b, have the


same valuation if and only if they generate the same ideal in AE . Let f(X) =∏h∈H(X − h(α)) be the minimal polynomial of α over L. We let τ acts on

the coefficients of f(X) ∈ AL[X] and we write τ(f)(X) =∏h∈H(X−τh(α))

the resulting polynomial. Each coefficient of τ(f)(X)− f(X) is of the form∑i≥1 ai(τ(βi) − βi) with ai ∈ AK . These coefficients are all divisible by

a = τ(β)−β. Plugging X = α in τ(f)(X)− f(X) we see that b = ±τ(f)(α)is divisible by a. Conversely, writing β = alα

l + · · ·+a0 = g(α) ∈ AK [α], weobtain that α is a zero of the polynomial g(X) − β. Therefore g(X) − β =f(X)p(X) with p(X) ∈ AE [X]. Now τ(g)(X) − τ(β) = g(X) − τ(β) =τ(f)(X)τ(p)(X). Plugging X = α in g(X)− τ(β) we obtain

−a = β − τ(β) = ±bτ(p)(α).

And the proposition follows.

Next one associates a real valued function ψE/K defined on [−1,∞) withthe filtration Gi as follows. For s ∈ [−1,∞) one puts

ψE/K(s) =∫ s

0

dt

(G0 : Gt)

with the convention that if t = −1, then (G0 : Gt) = (G : G0)−1, and for−1 < t ≤ 0, (G0 : Gt) = (G0 : G0)−1 = 1. The map f(t) := (G0 : Gt)−1

on [−1,∞) satisfies: f(−1) = (G : G0), f(t) = 1 for −1 < t ≤ 0 andf(t) = (G0 : Gn)−1 for n − 1 < t ≤ n and n ≥ 1. So f is a step functiondiscontinuous at 0, 1, 2, 3, · · · . Therefore ψE/K is continuous piecewise linearfunction. For an integer m, with 0 < m ≤ s ≤ m+ 1, we have

ψE/K(s) =∫ 1

0

dt

(G0 : Gt)+ · · ·+

∫ m−1

m

dt

(G0 : Gt)+∫ s

m

dt

(G0 : Gt).

This gives

ψE/K(s) =1g0

(g1 + g2 + · · ·+ gm + (s−m)gm+1), with gi = card(Gi).

The relation between this function and the integers iE/K(σ) is as follows.

Proposition 4.2.21. Let E/K be a Galois extension of local fields withGalois group G. Then one has

ψE/K(s) =1g0

∑σ∈G

inf{iE/K(σ), s+ 1} − 1.

Proof. Let θ(s) be the function on the right hand side. Both θ and ψE/Kare continuous and piecewise linear real functions defined on [−1,∞). Onehas ψE/K(0) = θ(0) = 0. On one hand for s ∈ (m,m + 1) with m ∈ Z, we


have inf{iE/K(σ), s+1} = s+1 if σ ∈ Gm+2 and it is the constant iE/K(σ)otherwise. Therefore for s ∈ (m,m+ 1),

θ′(s) =1g0card{σ ∈ G : iE/K(σ) ≥ m+ 2} =

1(G0 : Gm+1)

.

On the other hand it is clear that ψ′E/K(s) = 1(G0:Gm+1) for s ∈ (m,m+ 1).

Hence we deduce that we must have ψE/K(s) = θ(s).

Now via the function ψE/L one has the following result known as Her-brand’s theorem.

Theorem 4.2.22. ( Herbrand ) Let E/K be a Galois extension of localfields with G = Gal(E/K). Let H be normal in G, and let L = EH so thatL/K is Galois with Gal(L/K) = G/H. Lastly, let r = ψE/L(s). Then onehas

GsH/H = (G/H)r

Proof. To start let us first observe that ψE/L is an increasing function. Nextlet σ ∈ GsH/H and any τ ∈ G such that τ |L = σ. Then σ ∈ GsH/H ⇔iE/K(τ) ≥ s+ 1 for some τ. So, if we define j(σ) = supτ |L=σiE/K(τ), and byusing the fact that ψE/L is an increasing function, we can write

σ ∈ GsH/H ⇔ j(σ)− 1 ≥ s⇔ ψE/L(j(σ)− 1) ≥ ψE/L(s).

On the other hand, we have

σ ∈ (G/H)ψE/L(s) ⇔ iL/K(σ) ≥ ψE/L(s) + 1.

So, in order to prove GsH/H = (G/H)r, we are led to show the statement

ψE/L(j(σ)− 1) ≥ ψE/L(s)⇔ iL/K(σ)− 1 ≥ ψE/L(s),

that is we need to establish

ψE/L(j(σ)− 1) = iL/K(σ)− 1.

To this end we fix τ0 ∈ G with τ0|L = σ and j(σ) = iE/K(τ0). By proposition4.2.20 we have

iL/K(σ) =1

e(E/L)

∑τ |L=σ

iE/K(τ) =1

e(E/L)

∑h∈H

iE/K(τ0h).

Let α ∈ AE be a primitive element, i.e., AE = AK [α], then iE/K(τ0h) =vE(τ0(h(α)) − α) = vE(τ0(h(α) − α) + τ0(α) − α). Hence iE/K(τ0h) ≥inf{iE/K(h), j(σ)}. By definition of j(σ), we have iE/K(τ0h) ≤ j(σ) =iE/K(τ0). Whence if iE/K(h) ≥ j(σ), then iE/K(τ0h) = j(σ). Also in the case


iE/K(h) < j(σ), then the ultrametric inequality gives iE/K(τ0h) = iE/K(h).Thus we obtain iE/K(τ0h) = inf{iE/K(h), j(σ)}. Next, by proposition 4.2.21we have

ψE/L(s) + 1 =1

e(E/L)

∑h∈H

inf{iE/L(h), s+ 1}.

So after noting that iE/L(h) = iE/K(h), we deduce iL/K(σ)−1 = ψE/L(j(σ)−1). This is what we needed.

ψE/K is strictly increasing so it has an inverse say φE/K : [−1,∞) →[−1,∞). Then the upper numbering for the ramification groups is definedas follows.

Definition 4.2.23. For φE/K(r) = s, one puts Gr := Gs.

One can also write φE/K by means of an integral. Indeed on one handfor r and φE/K(r) non-integers, φ′E/K(r) = 1

ψ′E/K

(φE/K(r)). On the other

hand the function ω(s) =∫ s0 (G0 : Gt)dt, satisfies ω′(s) = 1

ψ′E/K

(φE/K(r))and

ω(0) = φE/K(0) = 0. Hence

φE/K(r) =∫ r

0(G0 : Gt)dt.

In a tower of fields these functions verify the following relations.

Proposition 4.2.24. Let L/K be a Galois subextension of E/K and H =Gal(E/L). Then

ψE/K = ψL/K ◦ ψE/L, φE/K = φE/L ◦ φL/K .

Proof. We take derivatives when it makes sense, that is at the points ofcontinuity of all the functions involved. The two functions on both sidesof the first relation take the same value zero at zero. We have ψ′E/K(s) =

1e(E/K)card(Gs).Also (ψL/K◦ψE/L(s))′ = card((G/H)t)card(Hs)

e(E/L)e(L/K) with t = ψE/L(s).

This is because we have ψ′E/L(s) = card(Hs)e(E/L) and ψ′L/K(t) = (G/H)t

e(L/K) . Hence itsuffices to see that card(Gs) = card((G/H)t)card(Hs). And this follows byHerbrand theorem, (G/H)t = GsH/H, and the surjective group homomor-phism ρ : Gs → GsH → Gs/H with kernel Gs

⋂H = Hs which induces the

isomorphism GsH/H ∼= Gs/Hs. Since ψE/K and φE/K are inverse of eachother, the second relation follows from the first.

Now putting all of this together one has the following.

Proposition 4.2.25. Let L/K be a Galois subextension of E/K with H =Gal(E/L). Then

GrH/H = (G/H)r.


Proof. By definition of the upper numbering GrH/H = GφE/K(r)H/H =(G/H)ψE/L(φE/K(r)) by Herbrand’s theorem. And since φE/L is the inverseof ψE/L we obtain ψE/L(φE/K(r)) = φL/K(r). Hence

GrH/H = (G/H)φL/K(r) = (G/H)r.

Thus we see that the lower numbering for the ramification groups is in-variant when changing the base field while the upper numbering is invariantwhen passing to a Galois subextension.

4.2.3 The group of norms

In this last part of this long chapter, we shall give the basic propertiesconcerning the norm group in a finite extension of local fields. This section isvery brief so the reader is invited to consult the references namely [10] or [15],for a satisfactory treatment of the norm group which plays a fundamentalrole in local class field theory as we will see in the next chapter.

So, let E/K be a Galois extension of local fields with G = Gal(E/K).Recall, the norm NE/K and the trace TrE/K are the group homomorphismsdefined by:

NE/K : E∗ → K∗

x 7→∏σ∈G

σ(x);

TrE/K : E → K

x 7→∑σ∈G

σ(x).

From their definitions we deduce the norm and the trace maps are continuoussince each σ ∈ Gal(E/K) is continuous. Particularly for the norm, thisimplies

Proposition 4.2.26. Let A∗E , A∗K be the unit groups. Then NE/K(A∗E)

is a compact subgroup of A∗K , hence closed in the compact Hausdorff groupA∗K . Furthermore NE/K(E∗) is closed in K∗.

Proof. A∗E is closed in the compact AE so it is also compact. From thecontinuity of the norm NE/K follows that NE/K(A∗E) is compact and henceclosed in A∗K . Next take a prime πE of E. Then we have E∗ =< πE > ×A∗E .Therefore NE/K(E∗) =< NE/K(πE) > ×NE/K(A∗E). Now < NE/K(πE) >is discrete in K∗ hence closed in K∗. Thus NE/K(E∗) is a product of groupsboth closed in K∗ so NE/K(E∗) is closed in K∗.


In the situation where E/K is not necessarily finite, one defines thenorm group N(E/K) and the unit norm group N(UE/K) as follows. ForL/K running over the finite subextensions of E/K, one puts N(E/K) =∩LNL/K(L∗), N(UE/K) = ∩LN(UL) where UL is the unit group of L.

Let E/K be finite unramified. We know that a prime πK of K is also aprime of E. Hence we can write the higher unit groups as follows: UE,i =1 + πiKAE , UK,i = 1 + πiKAK . For u = 1 + πiKa ∈ UE,i, we have

NE/K(u) =∏σ∈G

(1 + πiKσ(a)) ≡ 1 + πiKTrE/K(a) (mod πi+1K ). (4.1)

Thus

Lemma 4.2.27. For i ≥ 1 we have NE/K(UE,i) ⊂ UK,i.

Besides, we can define surjective homomorphisms

θE,i : UE,i → kE

1 + πiKa 7→ a;θK,i : UK,i → kK

1 + πiKa 7→ a.

By the congruence (4.1), one has

Proposition 4.2.28. The following diagram is commutative

UE,i

NE/K��

θE,i // kE

TrkE/kK��

UK,iθK,i // kK

Furthermore NE/K(UE,i) = UK,i for i ≥ 1.

Proof. We have θE,i(1 + πiKa) ≡ a (mod πK), and from the isomorphismGal(E/K) ∼= Gal(kE/kK), we see that TrE/K(x) = TrkE/kK (x). Combiningthis with the congruence for the norm makes the diagram commutative.We have the said equality from the surjectivity of the trace map TrkE/kK .To see this, recall that Gal(kE/kK) is cyclic generated by the Frobeniusautomorphism x 7→ xq with q = card(kK). So, the trace is the polynomialmap T (X) = Xqn−1

+ · · · + Xq + X on kE , where n = [kE : kK ] = [E :K]. Hence the kernel of TrkE/kK is the set of the zeros of T (X). Thuscard(ker(TrkE/kK )) ≤ qn−1, and hence card(im(TrkE/kK )) ≥ qn

qn−1 = q =card(kK).

Corollary 4.2.29. NE/K(A∗E) = A∗K .


Proof. Recall from proposition 3.2.18, p 30 that A∗E contains the (qn − 1)-th roots of unity and AK contains (q − 1)-th roots of unity. Then writeA∗E =< ζqn−1 > ×UE,1, where < ζqn−1 > is the cyclic group generated by aprimitive (qn − 1)-th root of unity ζqn−1. From proposition 4.2.28 we haveNE/K(UE,1) = UK,1 and NE/K(ζqn−1) is a primitive (q− 1)-th root of unityso that NE/K(< ζqn−1 >) =< ζq−1 > . Since A∗K =< ζq−1 > UK,1, we canconclude.

In the totally ramified situation we have the following useful criterion.

Proposition 4.2.30. An extension E/K is totally ramified if and only ifNE/K(E∗) contains a prime element.

Proof. We assume the extension to be finite; for the infinite case see [15].So if E/K is totally ramified, then we know that for a prime πE ∈ AENE/K(πE) is prime in K.Conversely, suppose thatNE/K(E∗) contains a prime πK . If E/K was not to-tally ramified, then we would have that E contains the unramified extensionofKur

n /K of degree n ≥ 2.We then obtainNE/K(E∗) ⊂ NKurn /K((Kur

n )∗) =<πnK > ×A∗Kur

nsince πK is a prime in Kur

n . A contradiction and so the propo-sition follows.

Chapter 5

Formal group law,Lubin-Tate extensions andLocal Class Field Theory

For the sake of giving the main theorem of the explicit description of the classfield theory of a local field, this chapter introduces the notion of a formalgroup law, with emphasis on Lubin-Tate formal group laws. By means ofthe latter we will define Lubin-Tate modules. We next give the constructionof the Lubin-Tate extensions which are obtained by adjoining torsion pointson Lubin-Tate modules to a complete unramified extension of a local field.And lastly local class field theory will follow. The main references here are[15], [25], [17] and [13].

5.1 Introduction

Let K be a local field with AK as ring of integers and mK the prime idealof AK . We fix an algebraic closure Kal of K and we write Kal, A, m forrespectively the completion of Kal with respect to the unique extension ofthe valuation of K to Kal, the valuation ring and the maximal ideal of Kal.Originally, explicit local class field for K relies on the notion of Lubin-Tateformal groups over AK , for the definition of formal groups see definition 5.2.1below. These Lubin-Tate groups give rise to an AK-module structure on m,see example 5.2.10 where the case K = Qp is illustrated. Then a tower oftotally ramified extensions of K generalising the cyclotomic extensions forQp is constructed by adjoining torsion points of the AK-module m to K.These generalised cyclotomic extensions are called Lubin-Tate extensions.This is Lubin-Tate theory as exposed in [16] and [21].

Now, let n be a positive integer and let L/K be the unique unramified ex-tension of degree n of K. In [22], relative Lubin-Tate groups are constructed,these encompass the Lubin-Tate groups as a special case. Similar as in the

55

56Formal group law, Lubin-Tate extensions and Local Class

Field Theory

classical case, in [15] and more recently in [25], a treatment of explicit localclass field theory for K is given by means of relative Lubin-Tate groups.Here, one constructs relative Lubin-Tate extensions of L by adjoining tor-sion points of the AK-module structure on m induced by relative Lubin-Tategroups. As this approach is a generalisation of the original treatment, wechoose to adopt it in our presentation of explicit local class field theory.

5.2 Relative Lubin-Tate formal group law

In this section, the concept of formal group law is introduced. We are mainlyinterested in a special class of formal group law that is the Lubin-Tate formalgroup law.

5.2.1 Generalities

Classically, by a group G one means a set G and a group law defined on G.By a formal group law, one is given a law without having a priori a set onwhich the law acts.

To start with we first recall some basic rules concerning formal powerseries which is the formalism in which formal group laws are defined. LetB = A[[X,Y ]] be the ring of formal power series in the variables X, Y overa ring A. Let F ∈ B. For a = (a1, a2) ∈ A2, recall that evaluating F at a isnot always defined. But for G(X,Y ), f(X) ∈ B with F (0) = 0, then we cansubstitute G ◦ f := G(f(X), f(Y )). On the subring A[[X]] of B, composingf(X), g(X) ∈ XA[[X]] defines a structure of semi-group ( not necessarilycommutative) with X as neutral element. f(X) ∈ XA[[X]] has an inverseg(X) ∈ XA[[X]] for composition that is f ◦ g = g ◦ f = X, if and only if thecoefficient of X in f is a unit.

For power series, the notation F ≡ G (mod (deg n)) signifies that thepower series F −G has all terms of degree greater or equal to n.

Definition 5.2.1. A formal group law over a ring A is a formal power seriesF (X,Y ) ∈ A[[X,Y ]] satisfying the following axioms:

(a) F (X,Y ) ≡ X + Y (mod (deg 2)),

(b) F (X,F (Y, Z)) = F (F (X,Y ), Z) (Associativity),

(c) F (X,Y ) = F (Y,X) (Commutativity).

Example 5.2.2. We have the additive formal group law given by Ga(X,Y ) :=X + Y and the multiplicative formal group law defined by Gm(X,Y ) :=X + Y +XY = (1 +X)(1 + Y )− 1.

From (a) and (b), one sees that a formal group law F (X,Y ) is of the formF (X,Y ) = X + Y +

∑i,j≥1X

iY j . Hence the equation F (X,Y ) = 0 can be

5.2 Relative Lubin-Tate formal group law 57

solved for Y ∈ XA[[X]]. Denote by ιF (X) its solution. For f, g ∈ XA[[X]],the law f+F g = F (f(X), g(X)), defines an abelian group on XA[[X]]. FromF (X, ιF (X)) = 0, we have f +F ιF (f) = F (f(X), ιF (f(X))) = 0, so ιF (f)is the inverse of f. When A is a complete valuation ring with maximal idealm, then for a, b ∈ m, the law a +F b = F (a, b) is well defined and makes(m, +F ) an abelian group.

We also have a notion of homomorphism of formal group laws. It isdefined as follows.

Definition 5.2.3. Let F,G ∈ A[[X,Y ]], be formal group laws. We say thata power series f(X) ∈ XA[[X]] is a homomorphism from F to G if:

f(F (X,Y )) = G(f(X), f(Y )).

It is an isomorphism if in addition there exists g(X) = f−1(X) ∈ XA[[X]]such that:

g(G(X,Y )) = F (g(X), g(Y )).

If f(X) admits an inverse g(X) for composition, then we have f(F (g(X),g(Y ))) = G(X,Y ) ⇔ g(G(X,Y )) = F (g(X), g(Y )). Therefore f is an iso-morphism of formal group laws if and only if the coefficient of X in f is aunit in A. Denote by HomA(F,G), the set of homomorphism from F to G.With the operation f +G g, HomA(F,G) is a subgroup of (XA[[X]],+G).Furthermore, with the multiplication f ◦ g, EndA(F ) := HomA(F, F ) has aring structure.Having at hand the basic language of formal group laws, we shall now turnto the study of Lubin-Tate formal group laws which are formal groups overa discrete valuation ring.

5.2.2 Relative Lubin-Tate formal group laws

Let E := Kur be the completion of the maximal unramified extension of alocal field K with vE the unique valuation on E extending vK of K. TheFrobenius automorphism ϕK ∈ Gal(Kur/K) is extended by continuity toE and we will denote it by ϕK . We write as usual AE , mE , AK , mK forthe valuation rings and maximal ideals of E, K respectively. Lastly letq = card(kK). We start by defining

Definition 5.2.4. Let π be a prime of E. A power series e(X) ∈ AE [[X]]is called a Lubin-Tate power series for π when the following conditions aresatisfied:

e(X) ≡ πX (mod (deg 2)), e(X) ≡ Xq (mod π).In particular a Lubin-Tate polynomial e(X) for π is of degree q and satis-fies these conditions. A Lubin-Tate polynomial is not necessarily a monicpolynomial.


Field Theory

Recall that ϕK acts on the ring AE [[X]] as follows. For f ∈ AE [[X]],fϕK is the power series obtained from the action of ϕK on the coefficients off. We next show that the Lubin-Tate power series arise as homomorphismsof certain formal group laws. In light of definition 5.2.3, for a power seriesF (X1, · · · , Xn) ∈ AE [[X1, · · · , Xn]] and a power series f(X) ∈ AE [[X]] wedefine F ◦ f := F (f(X1), · · · , f(Xn)) ∈ AE [[X1, · · · , Xn]].

Proposition 5.2.5. Let e(X), e′(X) ∈ AE [[X]] be Lubin-Tate power seriesfor the primes π, π′ respectively. Let L(X1, · · · , Xn) = a1X1 + · · ·+ anXn ∈AE [[X1, · · · , Xn]] such that π′L(X1, · · · , Xn) = πLϕK (X1, · · · , Xn), i.e.,π′ai = πaϕKi for all i. Then there exists a unique power series F (X1, · · · , Xn)∈ AE [[X1, · · · , Xn]] such that:

F (X1, · · · , Xn) ≡ L(X1, · · · , Xn) (mod deg 2) and e′ ◦ F = FϕK ◦ e.

Proof. A power series F (X1, · · · , Xn) with non constant term can be writtenas F (X1, · · · , Xn) =

∑∞r=1Hr(X1, · · · , Xn) with Hr(X1, · · · , Xn) ∈ AE [X1,

· · · , Xn], a homogeneous polynomial of degree r. Put Fm =∑m

r=1Hr. Weseek polynomials Fm,m ≥ 1, such that e′ ◦Fm ≡ FϕKm ◦e (mod deg m+1).Then, F = lim

m→∞Fm is the desired power series. For m = 1, we take F1 = L

as L satisfies the conditions by hypothesis. Suppose we have uniquely foundF1, · · · , Fm satisfying our requirement. Write Fm+1 = Fm +Hm+1, we havee′ ◦Fm+1 ≡ e′ ◦Fm + π′Hm+1 (mod degm+ 2) and also FϕKm+1 ◦ e ≡ F

ϕKm ◦

e+ πm+1HϕKm+1 (mod degm+ 2). Indeed, write e′(X) = π′X +

∑∞i=1 aiX

i.Then

e′ ◦ Fm+1 = π′Fm+1 +∞∑i=1

aiFim+1

= π′Fm + π′Hm+1 +∞∑i=1

ai(i∑

j=0

(i

j

)F i−jm Hj

m+1)

= π′Fm +∞∑i=0

aiFim + π′Hm+1 + ( terms of degree ≥ m+ 2).

For the second equation write e(X) = πX+∑∞

i=1 biXi and use the fact that

Hm+1 ◦ e = πm+1Hm+1 + (terms of degree ≥ m+ 2). Let Gm+1 = e′ ◦ Fm −FϕKm ◦ e. Then to satisfy our conditions we must have Gm+1 + π′Hm+1 −πm+1Hϕ

m+1 ≡ 0 (mod deg m + 2). On one hand, by definition of Gm+1,we have Gm+1 ≡ 0 (mod deg m + 1). On the other Fm(X1, · · · , Xn) =∑ai1,···,inX

i11 · · ·Xin

n , so that, F qm(X1, · · · , Xn) ≡∑aqi1,···,in(Xi1

i1)q · · · (Xin

n )q ≡FϕKm (Xq

1 , · · · , Xqn) (mod π′) since by definition of ϕK we have aϕK ≡ aq

(mod π′) ∀a ∈ AE . From e′(X) ≡ Xq (mod π′) we obtain e′ ◦ Fm ≡ F qm(mod π′) and similarly from e(X) ≡ Xq (mod π), we deduce that FϕKm ◦e ≡ FϕKm (Xq

1 , · · · , Xqn) (mod π′). Therefore we have Gm+1(X1, · · · , Xn) ≡

5.2 Relative Lubin-Tate formal group law 59

F qm(X1, · · · , Xn) − FϕKm (Xq1 , · · · , X

qn) ≡ 0 (mod π′), that is all the coeffi-

cients in Gm+1 are multiples of π′. So, let π′β, π′α, πm+1αϕK be the coeffi-cient of a monomial Xi1

1 · · ·Xinn of degree m+ 1 in Gm+1, π

′Hm+1, andπm+1HϕK

m+1 respectively. Then we must solve for α the equation π′β +π′α − πm+1αϕK = 0. This gives, with ω = π′−1πm+1, α = −β + ωαϕK =−β−ωβϕK +ωϕK+1αϕ

2K = · · · = −β−ωβϕK−ωϕK+2βϕ

2K−· · · . By the com-

pleteness of AE , we have α ∈ AE and it is unique. Indeed, if say α′ is anothersolution, then vE(α− α′) = vE(ω) + vE((α− α′)ϕE ) = vE(ω) + vE(α− α′).Since vE(ω) ≥ 1, we must have vE(α − α′) = ∞. This uniquely deter-mines Hm+1, and hence Fm+1 is uniquely determined which implies that Fis uniquely determined.

Corollary 5.2.6. Let e(X) ∈ AE [[X]] be a Lubin-Tate power series for aprime π. Then there exists a unique formal group law Fe over AE such thate(X) ∈ HomAE (Fe, F

ϕKe ).

Proof. Applying the proposition for π = π′, e = e′, L(X,Y ) = X+Y, we ob-tain a unique power series Fe(X,Y ) ∈ AE [[X,Y ]] such that e◦Fe = FϕKe ◦e.To see that Fe(X,Y ) is a formal group law, we have e ◦ (Fe(X,Fe(Y,Z))) =FϕKe (e(X), FϕKe (e(Y ), e(Z))) = (Fe(X,Fe(Y, Z))ϕK◦e. Similarly e◦(Fe(Fe(X,Y ), Z)) = (Fe(Fe(X,Y ), Z))ϕK◦e. Therefore Fe(X,Fe(Y, Z)) and Fe(Fe(X,Y ), Z) ∈ AE [[X,Y, Z]] satisfy the conditions of proposition 5.2.5 with π =π′, e = e′, L(X,Y, Z) = X+Y +Z. Hence they must be equal by the unique-ness of such a power series. The equality F (X,Y ) = F (Y,X) is obtained inthe same fashion.

Let AEπ,π′ = {a ∈ AE : π′a = πaϕK}, then from the proposition 5.2.5,for each a ∈ AEπ,π′ , there exists a unique power series [a]e,e′(X) ∈ AE [[X]]such that e′ ◦ [a]e,e′ = [a]ϕKe,e′ ◦ e, [a]e,e′(X) ≡ aX (mod deg 2). Let alsoFe, Fe′ be formal group laws over AE arising from Lubin-Tate power seriese(X), e′(X).

Proposition 5.2.7. Keeping the same notations as above, one has

1. [a]e,e′ ◦ Fe = Fe′ ◦ [a]e,e′ , i.e., [a]e,e′ ∈ HomAE (Fe, Fe′),

2. [a+ b]e,e′(X) = Fe([a]e,e′(X), [b]e,e′(X)),

3. If e′′ is a Lubin-Tate power series for a prime π′′ of AE , then [ab]e,e′′(X) =[a]e,e′([b]e′,e′′(X)),

4. If π is a prime of AK ( which is also a prime of AE ,) e(X) ∈ AE [[X]],the corresponding Lubin-Tate power series then π ∈ AEπ,π and [π]e :=[π]e,e = e(X).


Field Theory

Proof. These equalities are obtained by verifying in each case that the termssatisfy the conditions of proposition 5.2.5. For instance to check the firstequality, let L(X,Y ) = a(X + Y ). As [a]e,e′(X) ≡ aX (mod deg 2) andFe ≡ X + Y ≡ FϕKe (mod deg 2), we see that we have [a]e,e′ ◦ Fe ≡ FϕK ◦[a]e,e′ ≡ L (mod deg 2). Next, e′ ◦ [a]e,e′ ◦ Fe = [a]ϕKe,e′ ◦ e ◦ Fe = ([a]e,e′ ◦Fe)ϕK ◦ e. Similarly, one gets e′ ◦ Fe′ ◦ [a]e,e′ = (Fe′ ◦ [a]e,e′)ϕK ◦ e.

Corollary 5.2.8. We have an injective ring homomorphism

AK → EndAE (Fe), a 7→ [a]e := [a]e,e.

Proof. The first three equations in proposition 5.2.7 show that this is awell defined ring homomorphism and the property [a]e ≡ aX (mod deg 2)implies injectivity.

This leads to the following concept of formal Lubin-Tate module. It isprecisely defined as follows.

Definition 5.2.9. For any local field K a formal AK-module is a coupleF = (F (X,Y ), [.]) where F is a formal group law over AK and [.] is a ringhomomorphism:

AK → EndAK (F )a 7→ [a](X)

such that [a](X) ≡ aX (mod deg 2). When a prime π ∈ AK is fixed, thenit is called a formal Lubin-Tate module over AK for the prime π if in addition[π](X) ≡ Xq (mod π) with q = card(kK).

When F defines a group law on a set S as it is the case for the prime idealof a discrete valuation ring, then we can defines on S a module structure asfollows. For s ∈ S, a ∈ AK the action of a is defined by a.s := [a](s), in thecase where [a](s) converges.

Example 5.2.10. ( The case K = Qp) Let p be a rational prime numberas usual, take e(X) = (1 + X)p − 1 ∈ Zp[X], a Lubin-Tate polynomialfor p. E is the completion of the maximal unramified extension of Qp. LetGm(X,Y ) = (1 + X)(1 + Y ) − 1 ∈ Zp[X,Y ] be the multiplicative formalgroup law defined over Zp. Then

e ◦Gm(X,Y ) = (1 +X)p(1 + Y )p − 1 = Gm(e(X), e(Y )).

So, Gm is the Lubin-Tate formal group law over Zp corresponding to e. Forany n ≥ 1, let us define [n](X) := (1+X)n−1 ∈ Z[X]. It is also clear that wehave e◦[n] = [n]◦e. Now any a ∈ Zp is the limit of a sequence of integers {ai}.

One then defines [a](X) := (1 +X)a − 1 = limi→∞

(1 +X)ai − 1 =∞∑i=1

(a

i

)Xi

5.3 Relative Lubin-Tate extensions 61

with(ai

)= a(a−1)···(a−i+1)

i! ∈ Zp, the extension of the binomial coefficientto Zp. For each i we have that e ◦ [ai] = [ai] ◦ e and by continuity onehas that e ◦ [a] = [a] ◦ e. This says that [a] ∈ EndZp(Gm), and moreoverF = (Gm(X,Y ), [.]) defines a formal Lubin-Tate module for p. Let us denoteby A the completion of the ring of algebraic integers over Zp. Put mA = {a ∈A : v(a) > 0} with v the extension of the p-adic valuation to A. Then oneendowed mA with a structure of Zp-module by a.x = [a](x) = (1 + x)a − 1.This Zp-module has torsion points and for n ≥ 1 the pn-torsion points are :

{ζ : [pn](ζ − 1) = 0} = {ζ : ζpn

= 1}.

So, we see that adjoining pn-th root of unity to Qp amounts to adjoiningpn-torsion points of the Zp-module mA.

We shall next see that this construction can be carried in a more generalcontext. More precisely adjoining torsion points on Lubin-Tate modulesto a complete unramified extension L/K of a local field K is the mainconstruction that we shall study now.

5.3 Relative Lubin-Tate extensions

Let p be a rational prime number and let n ≥ 1 be an integer. To obtaintotally ramified extensions of Qp, one adjoins pn-th roots of unity. We sawthat these are roots of an Eisenstein polynomial over Qp and also that theGalois groups are isomorphic to (Z/pnZ)∗, see example 4.2.9. The content ofthis section is to see how the theory of formal group laws is used to generalizethat situation. Any local field in the sequel is of characteristic zero.

Let L/K be a complete unramified extension of a local field K andGalois. We can think of L as the completion of the maximal unramifiedextension of K or a finite extension of K. Let e(X) ∈ AL[X] be a Lubin-Tatepolynomial for a prime π of AL and let ϕK be the Frobenius of Gal(Kur/K).For m ≥ 1, form the polynomial em = eϕ

m−1K ◦ · · · ◦ eϕK ◦ e. Let hm(X) =

em(X)/em−1(X) = eϕm−1K (em−1(X))/em−1(X).

Lemma 5.3.1. hm(X) is an Eisenstein polynomial of degree (q − 1)qm−1.Furthermore hm(X) is irreducible and separable over L.

Proof. Let j(X) = eϕm−1K (X)/X, then as eϕ

m−1K (X) ≡ πϕ

m−1K X (mod deg 2),

we have j(X) ≡ πϕm−1K (mod deg 1). Since hm(X) = j(em−1(X)), the

claims follow by noting that j(X) ≡ Xq−1 (mod π) and recalling that incharacteristic zero irreducibility implies separability.

As a consequence we next obtain that the polynomial em(X) ∈ AL[X] isseparable. This follows from the separability of the Lubin-Tate polynomiale(X).


Field Theory

Lemma 5.3.2. e(X) is separable over AL. Furthermore em(X) is also sep-arable.

Proof. By definition of e(X), it is a polynomial of degree q satisfying e(X) ≡Xq (mod π) and e(X) ≡ πX (mod deg 2). Hence, e(X)/X is an Eisen-stein polynomial, so it is irreducible and since we are in characteristic zero itis separable. By e(X) = Xe(X)/X, we deduce that e(X) is separable. Weget the separability of em(X) by induction on m. We just saw the case m = 1as e1 = e. Suppose then em−1 is separable. From em(X) = em−1(X)hm(X),one deduces the separability of em(X) since the product of distinct polyno-mials all separable is separable.

Set µe,m = {α : em(α) = 0}. Put L′ = L(µe,m), the splitting field ofem(X).

Definition 5.3.3. The extension L′/L is called a relative Lubin-Tate exten-sion.

Proposition 5.3.4. 1. The set µe,m is an AK-module by +Fe and [.]e.For any α ∈ µe,m, we have an AK-module homomorphism:

ψ : AK → µe,m, a 7→ [a]e(α).

It induces an isomorphism of AK-modules:

AK/mmK → µe,m

a 7→ [a]e(α)

for α ∈ µe,m \ µe,m−1.

2. L′ = L(α) where α is a zero of the Eisenstein polynomial hm(X) ∈AL[X] so that L′/L is totally ramified of degree (q − 1)qm−1.

Proof. 1. It is easy to see µe,1 ⊂ mL′ . Any α ∈ µe,m−i\µe,m−i−1 is a zero ofthe Eisenstein polynomial hm−i(X) ∈ AL[X] for 0 ≤ i ≤ m− 2. Thusα ∈ mL′ . Writing µe,m =

⋃m−2i=0 µe,m−i \ µe,m−i−1

⋃µe,1, we obtain

µe,m ⊂ mL′ . Therefore, [a]e(α), Fe([a]e(α), [a]e(β)) ∈ µe,m for α, β ∈µe,m and a ∈ AK . Indeed by induction one has em ◦ [a]e = [a]e ◦ emand em ◦ Fe = Fe ◦ em. Hence (em ◦ [a]e)(α) = [a]e(em(α)) = [a]e(0) =0. Similarly em ◦ Fe([a]e(α), [a]e(β)) = Fe(em([a]e(α), em([a]e(β))) =Fe(0, 0) = 0.Now, from proposition 5.2.5, we obtain e(X) = [π]e(X). Thus em(X) =[πϕ

m−1K +···+ϕK+1]e(X) = [uπmK ]e(X) with u ∈ A∗L. Writing em(X) =

[u]e(X) ◦ [πmK ]e(X), one sees that: em(α) = 0 ⇐⇒ [πmK ]e(α) = 0. Thismeans that any α ∈ µem \ µe,m−1 is exactly annihilated by [πmK ]e(X),that is to say [πk]e(α) 6= 0 for 1 ≤ k ≤ m − 1. This implies that the


induced homomorphism has kernel mmK and hence the isomorphism

follows. This ends the proof of the first part of the proposition.

2. We have L′ = L(µe,m). As µe,m = {[a]e(α) : a ∈ AK}, for anyα ∈ µe,m \ µe,m−1, it follows that L′ = L(α), with α a root of theEisenstein polynomial hm(X). Therefore L′/L is totally ramified ofdegree deg(hm) = (q − 1)qm−1.

Let α ∈ µe,m \ µe,m−1 be fixed. The Galois group Gal(L′/L) is com-puted as follows. By Galois theory we have Gal(L′/L) ⊂ Aut(µe,m) ∼=Aut(AK/mm

K) ∼= (AK/mmK)∗ where [u]e( . ) ∈ Aut(µe,m) maps to u ∈ (AK/mm

K)∗.This homomorphism is clearly onto and since card(Gal(L′/L)) = (q −1)qm−1 = card((AK/mm

K)∗), one has an isomorphism:

ρe,m : Gal(L′/L) → (AK/mmK)∗

(σ : σ(α) = [u]e(α)) 7→ u.

This isomorphism does not depend on the choice of a primitive elementα ∈ µe,m\µe,m−1. Indeed, let σ(α) = [u]e(α) and α′ = [u′]e(α) be a conjugateof α. Note that [u]e(X) ∈ AL[[X]] and hence σ([u′]e(α)) = [u′]e(σ(α)) =[u′]e([u]e(α)) = [u]e([u′]e(α)). We have proved the following

Proposition 5.3.5. The Galois group Gal(L′/L) is isomorphic to (AK/mmK)∗.

The isomorphism being given by ρe,m : Gal(L′/L)→ (AK/mmK)∗; [u]e(.) 7→ u

(mod mm) and is independent of α.

5.3.1 Isomorphism of Lubin-Tate extensions

Let E = Kur, be the completion of the maximal unramified extension ofK. Let e(X), e′(X) ∈ AE [[X]] be Lubin-Tate power series for the primesπ, π′ ∈ AE respectively. Let Fe(X,Y ), Fe′(X,Y ) ∈ AE [[X,Y ]] be the formalgroup laws corresponding to e(X), e′(X) respectively. For a ∈ AEπ,π′ = {a ∈AE : π′a = πaϕK}, consider [a]e,e′(X) ∈ AE [[X]] ∈ Hom(Fe, Fe′). Finally,ϕK denotes the extension of the Frobenius automorphism of Gal(Kur/K)to E.

We first prove that in this setting, the Lubin-Tate formal group lawsFe, Fe′ over E are in fact isomorphic.

Lemma 5.3.6. The map

ψ : A∗E → A∗E

u 7→ uϕK/u

is surjective.


Field Theory

Proof. As A∗E ∼= lim←m

(AE/mm+1E )∗, it suffices to verify that for v ∈ AE and

for all m ≥ 0, there exists um such that uϕKm /um ≡ v (mod πm+1E ). When

m = 0, then (AE/mE)∗ ∼= F∗q . Thus ψ(u) = uϕK/u = uq−1, and this is asurjective map as we are in the algebraic closure of Fp. Suppose then thatwe have found um such that v/(uϕKm /um) = 1+απm+1

K with πK a prime of Kand α ∈ AE . On AE/mE

∼= Fq, the map u 7→ ¯uϕK − u = uq − u is surjectiveso that there exists β ∈ AE such that βϕK − β ≡ α (mod mE). Then,setting um+1 = um(1 + βπm+1), gives uϕKm+1/um+1 ≡ v (mod mm+2

E ).

This gives

Proposition 5.3.7. Let e, e′, Fe, Fe′ be as above. Then a ∈ AEπ,π′⋂A∗E gives

rise to an isomorphism of the Lubin-Tate formal group laws over E :

[a]e,e′ : Fe ∼= Fe′ .

Proof. We know that [a]e,e′ ≡ aX (mod deg 2). Hence [a]e,e′ ∈ XAE [[X]]is invertible, and so the homomorphism [a]e,e′ is an isomorphism over E.

The extension L/K is still complete unramified inside E. As a prime inL is also a prime in E, a Lubin-Tate polynomial over AL is also a Lubin-Tatepolynomial over AE .

Next, take e(X), e′(X) ∈ AL[X], Lubin-Tate polynomials for the primesπ, π′ ∈ L and [a]e,e′ ∈ HomE(Fe, Fe′). An isomorphism [a]e,e′ : Fe ∼= Fe′ overE gives rise to an isomorphism of AK-modules µe,m, µe′,m as follows. Fora ∈ AEπ,π′

⋂A∗E , we have [a]e,e′(X) ∈ AE [[X]] such that [a]ϕKe,e′ ◦e = e′ ◦ [a]e,e′ .

Hence for m ≥ 1, we get e′m ◦ [a]e,e′ = [a]ϕmKe,e′ ◦ em. Then for α ∈ µe,m, we see

that [a]e,e′(α) ∈ µe′,m. Therefore, we have a map:

[a]e,e′ : µe,m → µe′,m

α 7→ [a]e,e′(α).

From the properties of [a]e,e′(X), one deduces that this is a homomorphismof AK-modules and hence that this is an isomorphism since a is a unit. Ifin addition we suppose that [a]e,e′(X) ∈ AL[[X]], then we have L(µe,m) =L(µe′,m) and ρe,m = ρe′,m. Indeed, because L is complete then L(µe,m) iscomplete and so µe′,m = [a]e,e′(µe,m) ⊂ L(µe,m) so that L(µe′,m) ⊂ L(µe,m).The reverse inclusion follows similarly since µe,m = [a−1]e′,e(µe′,m). Fromproposition 5.3.5 and for α ∈ µe,m, we have an isomorphism:

ρe,m : Gal(L(µe,m)/L) → (AK/mmK)∗

(α 7→ [u]e,e′(α)) 7→ u (mod mmK).

Hence, the automorphism [u]e,e′(α) maps to [u.a]e,e′(α) under the isomor-phism [a]e,e′(.) : µe,m ∼= µe′,m. From [a.u]e,e′(α) = [u]e′([a]e,e′(α)), one sees


that ρe,m = ρe′,m. Summarizing this discussion we have

Proposition 5.3.8. Let L/K be a complete unramified extension of a localfield K. Let e(X), e′(X) ∈ AL[X] be Lubin-Tate polynomials for the primesπ, π′ ∈ L respectively. Let a ∈ AEπ,π′

⋂A∗E so that [a]e,e′(X) ∈ AE [[X]]

defines an isomorphism: Fe ∼= Fe′ of formal groups law over AE . Then wehave an isomorphism of AK-modules

1. [a]e,e′(.) : µe,m ∼= µe′,m, and

2. Furthermore if [a]e,e′ ∈ AL[[X]], then L(µe,m) = L(µe′,m) and ρe,m =ρe′,m.

We now make the hypothesis that the extension L/K is finite insideE = Kur. The following result gives a criterion that allows one to decidewhether [a]e,e′(X) ∈ AL[[X]] or not. We need the following. For a Lubin-Tate power series e(X) ∈ AL[[X]], the map :

· ◦ e : AL[[X]] → AL[[X]]h 7→ h ◦ e

is well defined since e(0) = 0.

Lemma 5.3.9. This map is an injection.

Proof. Let π be a prime of L and q = card(kL). The injectivity follows fromthe statement h◦e ≡ 0 (mod πm)⇒ h ≡ 0 (mod πm) for m ≥ 0. One seesthis by induction on m. For m = 0, this is clear. Suppose then that this istrue up to m−1. If h◦e = πmg, then by induction hypothesis h = πm−1h′ sothat h′ ◦ e = πg. Reduction modulo π, leads h′(Xq) ≡ (h′(X))q ≡ h′(X) ≡ 0(mod π). This gives h(X) ≡ 0 (mod πm). Therefore if (h1 − h2) ◦ e ≡ 0(mod πm) for m ≥ 0, one deduces that we must have h1 = h2 by takinglimit.

Then we can state

Proposition 5.3.10. Let L/K be a finite unramified extension of degreen. Let e(X), e′(X) be Lubin-Tate polynomials for the primes π, π′ ∈ Lrespectively. Let [a]e,e′(X) ∈ IsomAE (Fe, Fe′) and let ϕK be the Frobe-nius in Gal(Kur/K). Then [a]ϕ

nKe,e′ = [a]e,e′ ◦ [NL/K(π′/π)]e. Furthermore,

if NL/K(π) = NL/K(π′), then [a]e,e′(X) ∈ AL[[X]].

Proof. By definition of en, one has en ≡ πϕn−1K · · ·πϕKπX ≡ NK(π)X

(mod deg 2). On one hand, since ϕnK is the identity on L, one gets e ◦ en =en ◦ e. On the other, [NL/K(π)]e clearly satisfies: [NL/K(π)]e ≡ NL/K(π)X(mod deg2) and e◦[NL/K(π)]e = [NL/K(π)]e◦e.Hence, en(X) = [NL/K(π)]e(X)by proposition 5.2.5. Similarly, one gets e′n = [NL/K(π′)]e′ . As e′ ◦ [a]e,e′ =


Field Theory

[a]ϕKe,e′ ◦ e, we obtain e′n ◦ [a]e,e′ = [a]ϕnK ◦ en. Thus [a]ϕ

nK ◦ en = [NL/K(π′)]e′ ◦

[a] = [a.NL/K(π′)]e,e′ = [a]e,e′ ◦ [NL/K(π′)]e = [a]e,e′ ◦ [NL/K(π′/π)]e ◦en. Now to conclude the equality use the above lemma successively fore, eϕK , · · · , eϕ

n−1K . Next if NL/K(π′) = NL/K(π), then one has [a]ϕ

nKe,e′ = [a]e,e′ ,

this means that [a]e,e′(X) ∈ AL[X].

From the above discussion, we see that when L/K is a finite unramifiedextension, then the Lubin-Tate extensions L(µe,m) are dependent only onNL/K(π) = x ∈ K∗ with π a prime in L defining the Lubin-Tate polynomiale(X) ∈ AL[X]. So, let us denote L(µe,m) by Km

x . The maps ρe,m dependalso only on NL/K(π) and so we will write ρm for ρe,m.

Let [L : K] = n. Let πK ∈ K be a prime. It is also a prime in L.Now any prime π of L can be written as π = uπK with u ∈ A∗L. HenceNL/K(π) = NL/K(u)πnK . Thus we have vK(NL/K(π)) = n. So if x ∈ K∗ withvK(x) = n then we have x = yNL/K(π) with y ∈ A∗K . Next as NL/K mapsA∗L onto A∗K since L/K is unramified, there is w ∈ A∗L with NL/K(w) = y.Whence x = NL/K(wπ). This means that if x ∈ K∗ with vK(x) = n, thenthere exists a prime π′ ∈ L such that x = NL/K(π′).

For m,m′ ≥ 1, say m ≤ m′, then em | em′ and hence Kmx ⊂ Km′

x . Thus⋃m≥1K

mx is a totally ramified extension of L. It is denoted by Kram

x . Thecanonical projection Gal(Km′

x /L)→ Gal(Kmx /L) gives rise to the projective

system (Gal(Kmx /L),m ∈ N). Let us recall also that Gal(Km

x ) ∼= (AK/mmK)∗.

This gives rise to an isomorphism:

Gal(Kramx /L) ∼= lim

←m

Gal(Kmx /L) ∼= lim

←m

(AK/mmK)∗ ∼= A∗K .

For the maximal unramified extension Kur =⋃n≥1K

urn , where Kur

n is theunique unramified extension of degree n of K, we have Gal(Kur

n /K) ∼=Gal(kKur

n/kK) ∼= Z/nZ. Then if n′ | n, we have restriction: Gal(Kur

n /K)→Gal(Kur

n′ /K). Hence, one has Gal(Kur/K) ∼= lim←n

Z/nZ = Z, the isomor-

phism being given by ϕK ↔ 1, where ϕK is the Frobenius of Gal(Kur/K).From the equality Lur = Kur, one has Gal(Kur/L) = Gal(Lur/L) = Z, theisomorphism is given by ϕnK 7→ 1 with n = [L : K].

Now define the extension KLTx = Kram

x Kur of L. This is an abelianextension of L. Since Kram

x /L is totally ramified and Kur/L is unramified,one sees that Kram

⋂Kur = L. Therefore by Galois theory we have

Gal(KLTx /L) ∼= Gal(Kram/L) × Gal(Kur/L) ∼= A∗K × Z

(α 7→ [u](α), ϕbL) 7→ (u, b).

Then a map NL/K(L∗) = A∗K× < x >→ Gal(KLTx /K) is defined as follows.

Definition 5.3.11. (Artin maps) The Artin map associated to x and


denoted by ArtxK is the map:

ArtxK : NL/K(L∗)→ Gal(KLTx /K)

with ArtxK(uxb) acting as [u−1] on each Kmx and like ϕbL on Kur where ϕL

is the Frobenius in Gal(Lur/L) which is ϕnK with n = [L : K] and ϕK theFrobenius in Gal(Kur/K).

Remark 5.3.12. Using [u−1] and not [u] is precisely because we want it tobe independent of x.

In what follows, we will see that the extension Kmx K

ur is independentof the choice of x with vK(x) = n = [L : K] and that the Artin mapArtxK is independent of x and is a restriction of a certain map ArtK : K∗ →Gal(KLT /K). On the way to these statements we need the following lemma.

Lemma 5.3.13. Let K be a local field with Kal as algebraic closure. Let Sbe any extension of K inside Kal. Lastly, let E′, E′′ be algebraic extensionsof S. Then the following holds

1. If E′/S is finite then E′S = E′,

2. S⋂E′ = S,

3. E′ = E′′ ⇒ E′ = E′′.

Proof. 1. Let E′/S be finite. Therefore E′S/S is finite. Hence E′S iscomplete in Kal, the completion of the algebraic closure of K, byrecalling that a finite extension of a complete field is also complete.Then E′S = E′S = E′S = E′.

2. To prove S ∩E′ = S, we can assume that the extension E′/S is finiteGalois. Indeed, we can take the normal closure R of E′/S and verifythat R ∩ S = S. So, let E′/S be Galois so that E′S/S is also Galois.Now any automorphism σ ∈ Gal(E′/S) can be extended uniquely bycontinuity to an automorphism σ ∈ Gal(E′/S). Therefore

[E′ : S] ≤ [E′ : S] = [E′S : S] = [E′ : S ∩ E′].

This means that S ∩ E′ = S. Now if E′/S is an infinite extensionwe can write E′ = ∪Ei with Ei/S finite so that S

⋂Ei = S. Then

S⋂E′ =

⋃(S ∩ Ei) = S.

3. Applying the second part to E′E′′, E′, andE′′, one has E′E′′⋂E′ =

E′, andE′E′′⋂E′′ = E′′. Thus, the condition E′ = E′′ implies E′ =

E′′.


Field Theory

Then

Theorem 5.3.14. 1. The fields Kmx K

ur for m ≥ 1 and hence theirunion KLT

x are independent of the choice of x ∈ mK∩K∗ with vK(x) =n.

2. The Artin map ArtxK is independent of x with vK(x) = n. Furthermoreif vK(x) = 1, then ArtxK =: ArtK : K∗ → Gal(KLT /K) is such thatfor y ∈ K∗ with vK(y) = n, we have ArtyK = ArtK |NL/K(L∗) with L/Kthe finite unramified extension of degree n.

Proof. 1. Let x, x′ ∈ K∗ with vK(x) = vK(x′) = n, and let π, π′ beprimes in L such that NL/K(π) = x and NL/K(π′) = x′. We consideralso the Lubin-Tate polynomials e(X), e′(X) ∈ AL[X] for π and π′

respectively. We write also µe,m, µe′,m for the set of roots of the poly-nomials em(X), e′m(X) as defined before lemma 5.3.1, p 61. By propo-sition 5.3.7, p 64, there exists an isomorphism [a]e,e′ : µe,m ∼= µe′,m,

so that µe′,m = [a]e,e′(µe,m) where [a]e,e′(X) ∈ Kur[[X]]. We haveKmx′ = L(µe′,m) so that Km

x′ Kur = Kur(µe′,m). Similarly one obtains

that Kmx K

ur = Kur(µe,m). Then since we have µe′,m = [a]e,e′(µe,m)with [a]e,e′(X) ∈ Kur[X], one deduces that Kur(µe′,m) = Kur(µe,m).Now Km

x′Kur/Kur is a finite extension so by applying the first part

of lemma 5.3.13 we have Kmx′K

urKur = ˆ(Kmx′K

ur) = Kur(µe′,m). Sim-ilarly we have Km

x KurKur = ˆ(Km

x Kur) = Kur(µe,m). So, the fields

Kmx and Km

x′ have the same completion. From the last part of lemma5.3.13, we obtain Km

x Kur = Km

x′Kur. This means that Km

x = Kmx′ .

2. We are still in the same setting. We need to show that ArtxK(x′) =Artx

′K(x′). By definition ArtxK(x′) acts as ϕnK on Kur where ϕK the

Frobenius of Gal(Kur/K). So, we have that ArtxK = Artx′K on Kur.

Next we prove that ArtxK(x′) is the identity on Kramx′ . To this end,

let [a]e,e′(X) =∑aiX

i ∈ Kur[[X]] and write x′ = x′

x x = ux. Then,by definition ArtxK(x′) acts as α 7→ [u−1]e(α) with α ∈ µe,m \ µe,m−1.Hence, we haveArtxK(x′)([a]e,e′(α)) =

∑ArtxK(x′)(ai)(ArtxK(x′)(α))i =∑

aϕnKi ([u−1]e(α))i = ([a]ϕ

nKe,e′ ◦ [u−1]e)(α). From proposition 5.3.10, we

obtain

ArtxK(x′)([a]e,e′(α)) = ([a]e,e′ ◦ [u]e ◦ [u−1]e)(α) = [a]e,e′(α).

Thus, ArtxK(x′) is the identity on Kramx′ since it is so on µe′,m for

all m ≥ 1. Similarly for any x′′ ∈ K∗ with vK(x′′) = n, we obtainArtx

′K(x′′) = Artx

′′K (x′′) = ArtxK(x′′), that is ArtxK = Artx

′K . Now, if

vK(x) = 1, we have Kmx = K(µe,m). Then for an unramified extension

L/K of degree n = vK(xn), one gets Kmxn = L(µe,m) = Km

x L and fromthe definition of the Artin maps we have Artx

n

K = ArtxK |NL/K(L∗).

5.4 Local class field theory 69

Having this at hand we make the following definition.

Definition 5.3.15. 1. Let π be a prime of K and let ϕK be the Frobe-nius automorphism in Gal(Kur/K). The Artin map of K is the homo-morphism

ArtK : K∗ → Gal(KLT /K)uπl 7→ [u−1]eϕlK .

2. Let E/K be an algebraic extension that contains Kur. The Weil groupW (E/K) of the extension E/K is

W (E/K) = {σ ∈ Gal(E/K) : σ|Kur ∈ ϕZK}.

Then theorem 5.3.14 has the following corollary.

Corollary 5.3.16. If x ∈ K∗, then σ = ArtK(x) ∈ Gal(KLT /K) is char-acterized by σ|Kur = ϕ

vK(x)K and σ|Kram

x= 1. By the Artin map we have

ArtK : K∗ →W (KLT /K) isomorphically.

Proof. The first statement is just a rewriting of the definition of the Artinmap. The last statement is clear from the definition of the Weil group andthe fact that for x ∈ K∗, ArtK(x)|Kur = ϕ

vK(x)K .

Next come the classical theorems for local class field theory.

5.4 Local class field theory

Let K be local field, we prove that by the Artin map ArtK : K∗ →Gal(KLT /K), one classifies the abelian extensions of K. To this end, wefirst establish that the extension KLT is in fact the abelian closure of K.This is the local Kronecker-Weber theorem for K.

5.4.1 The local Kronecker-Weber theorem

We follow here a classical argument that uses the Hasse-Arf theorem . So,let us start by recalling the statement of the said theorem.

Let F/K be a Galois extension of local fields with Galois group G. LetGi be the ramification groups in the lower numbering and the let ψF/K(s) =1g0

(g1 + · · ·+ gm + (s−m)) with s ∈ [−1,∞), 0 < m ≤ s ≤ m+ 1 and gi =card(Gi), be the real valued function associated with the groups Gi as de-fined in 4.2.15, p 46. Recall also that the ramification groups in uppernumber numbering are Gm = Gψ−1

F/K(m). Then the Hasse-Arf theorem reads

as follows.


Field Theory

Theorem 5.4.1. (Hasse-Arf) If G is abelian, n ∈ Z≥0 and Gn 6= Gn+1, thenψF/K(n) ∈ Z≥0.

Proof. See [20], or [25].

We also need the following result concerning the computation of theramification groups in upper numbering of the totally ramified extensionsKmx /K for m ≥ 1.

Proposition 5.4.2. Let G be the Galois group of the extension Kmx /K for

m ≥ 1. Then

1. G0 = G, G1 = G2 = · · · = Gq−1, Gq = · · · = Gq2−1, · · · , Gqm−1 = 1;

2. G0 = G0, G1 = Gq−1, · · · , Gm = Gqm−1 = 1.

Proof. See [17, pp 31-32].

We will make use as well of the

Proposition 5.4.3. 1. Let G be the Galois group of a totally ramifiedabelian extension of local fields F/K, and let q = card(kK) where kKis the residue field of K. Then (G : Gm) divides (q− 1)qm−1 with m apositive integer.

2. Let K ′/K and K ′′/K be two Galois extensions with K ′K ′′/K totallyramified. If Gal(K ′/K)m = Gal(K ′′/K)m = 1, then Gal(K ′K ′′/K)m

= 1.

Proof. 1. We write ψ for ψF/K . Recall first that we have embeddingsG0/G1 ↪→ k∗K , and Gi/Gi+1 ↪→ kK for i ≥ 1, see 4.2.17, p 47. AsGm = Gψ−1(m), we know that for n ∈ Z≥0 if n− 1 < ψ−1(m) ≤ n, i.e.,ψ(n − 1) < m ≤ ψ(n) (ψ is an increasing function), then Gm = Gn.As (G : Gn) = (G : G1)(G1 : G2) · · · (Gn−1 : Gn), by the Hasse-Arftheorem, if 1 ≤ i ≤ n, thenGi−1 6= Gi occurs only when ψ(i−1) ∈ Z≥0.(G : G1) divides q− 1 and for i ≥ 2 we have that Gi−1 6= Gi occurs atmost m − 1 times as ψ(i − 1) ≤ ψ(n − 1) < m. Hence as (Gi−1 : Gi)divides q we see that (G : Gm) divides (q − 1)qm−1.

2. Put S = Gal(K ′K ′′/K) and letH = Gal(K ′K ′′/K ′′) so thatGal(K ′′/K)= S/H. We know that SmH/H = (S/H)m = 1, hence Sm ⊂ H =Gal(K ′K ′′/K ′′). Similarly Sm ⊂ Gal(K ′K ′′/K ′). Thus Sm ⊂ Gal(K ′K ′′/K ′)

⋂Gal(K ′K ′′/K ′′) = 1.

Let now π be a prime of a local field K and consider the extensionKramπ =

⋃m≥1K

mπ .


Lemma 5.4.4. Kramπ is a maximal totally ramified extension of K inside

Kab.

Proof. Let E be an intermediate field of Kab/Kramπ and totally ramified over

K. As each finite subextension E′ of E/K is totally ramified, we have to showthat E′ ⊂ Km

π for some m ≥ 1. So, let E′/K as said with G = Gal(E′/K).For m large enough we have Gm = {id}. Combining this with proposition5.4.2, 2., and 5.4.3, 2., one gets Gal(E′Km

π )m = 1. From proposition 5.4.3,1., we have [E′Km

π : K] | (q − 1)qm−1 = [Kmπ : K]. Therefore E′ ⊂ Km

π .

Then we have the following statement.

Theorem 5.4.5. (Local Kronecker-Weber theorem ) Every abelianextension of a local field K lies inside KLT , i.e., KLT = Kab.

Proof. Let ϕK be the Frobenius element ofGal(Kur/K), and let σ ∈ Gal(Kab

/K) be any extension of ϕK . Set F the fixed field of σ. By definitionF⋂Kur = K, hence F/K is totally ramified. The rule Gal(Kab/F ) →

Z, σ 7→ 1 gives an isomorphism Gal(Kab/F ) ∼= Z. In fact, it is easy tosee that Gal(Kab/F ) ∼= Gal(Kur/K) by mapping σ to ϕK . Since Z hasa unique closed subgroup of index n, namely nZ, there exits only one in-termediate field of Kab/F of degree n. Since F

⋂Kur = F

⋂Kurn where

Kurn is the unique unramified extension of degree n over K, FKur

n /F is thesubextension of degree n over F. Hence we obtain Kab =

⋃[F ′:F ]<∞ F

′/F =⋃n≥1 FK

urn = FKur. One the other hand, let π be a prime of K. By def-

inition ArtK(π)|Kur = ϕK , and ArtK(π)|Kramπ

= id. Thus one sees thatKramπ ⊂ F, which implies F = Kram

π by the maximality of Kramπ .

5.4.2 The theorems of local class field theory

For a local field K, we have ArtK : K∗ → Gal(Kab/K). Let K ′/K be anextension of local fields. Obsverve that Kab ⊂ K ′ab so that the restrictionmap res : Gal(K ′ab/K ′) → Gal(Kab/K) is well defined. We shall next seehow the Artin map behaves with respect to a change of the ground field fromK to K ′. We begin with the following result concerning the norm group ofthe extension Km

x /K. So let L/K be a finite unramified extension and π aprime of L with x = NL/K(π) so that vK(x) = [L : K]. Also for an infiniteextension E/K we write N(E/K) to denote ∩NEi/K(E∗i ) with Ei/K finitesubextensions of E/K.

Lemma 5.4.6. Let K be a local field and let x ∈ K∗ as above. ThenNKm

x /K((Km

x )∗) = (1 + mm

K)× < x > . Furthermore, if E/K is a totallyramified extension and E ⊃ Kram

x , then N(E/K) =< x > .

Proof. See [25, p 11].


Field Theory

Theorem 5.4.7. Let K ′/K be an extension of local fields. The followingdiagram is commutative:

K ′∗

NK′/K

��

ArtK′// Gal(K ′ab/K ′)

res

��K∗

ArtK// Gal(Kab/K)

Proof. Let π′ be a prime of K ′, and let ϕ′ be the Frobenius of K ′ur. By defini-tion ofArtK′(π′) ∈ Gal(K ′ab/K ′) it verifiesArtK′(π′)|K′ur = ϕ′K′ and ArtK′(π′)|K′ram

π′= 1. Hence the fixed field of ArtK′(π′) is K ′ramπ . As K ′ramπ′ /K ′ is

totally ramified, from lemma 5.4.6, N(K ′ramπ′ /K ′) =< π′ > . Therefore bythe transitivity of the norm we have N(K ′ramπ′ /K) =< NK′/K(π′) > . Thenset π = Art−1

K (ArtK′(π′)|Kab) ∈ K∗ and let L be the maximal unramifiedsubextension of K ′/K i.e., L = K ′

⋂Kur. As K ′ramπ′ /L is totally ramified

and Kramπ ⊂ K ′ramπ′ (by the maximality of K ′ramπ′ ), from lemma 5.4.6 we

get N(K ′ramπ′ /K) =< π > . Hence as π and NK′/K(π′) generate the samegroup, they must be associate. By changing the prime π if necessary wemay write π = NK′K(π′). Thus ArtK′(π′)|Kab = ArtK(NK′/K(π′)). Nowuse the fact that as a group K ′∗ is generated by the primes to conclude thatArtK′ ◦ res = ArtK ◦NK′/K .

By theorem 5.4.7, we see that if K ′/K is a finite abelian extensionthen ArtK(NK′/KK

′∗)|K′ = 1. We also know that for a prime π ∈ K∗

we have ArtK(π)|Kur = ϕK with ϕK the Frobenius of Gal(Kur/K) andArtK(π)|Kram

π

= 1. Now suppose that we have ψ : K∗ → Gal(Kab/K) another homomor-phism satisfying all these properties. Let π ∈ K be a prime and considerthe extension Kram

π . Then π ∈ N(Kramπ /K) so that ArtK(π)|Kram

π= 1

by definition. This means that ArtK(π)|Kur = ϕK = ψ(π)|Kur and alsoArtK(π)|Kram

π= 1 = ψ(π)|Kram

π. Therefore, one has ArtK(π) = ψ(π). Since

K∗ is generated by the primes as a group we conclude that ArtK = ψ.In other words the homomorphism ArtK : K∗ → Gal(Kab/K) is uniquelycharacterized by:

ArtK(NK′/KK′∗)|K′ = 1 and ArtK(π)|Kur = ϕK .

We also have

Theorem 5.4.8. Let K ′/K be a finite extension of local fields. Then wehave an isomorphism

K∗/NK′/K(K ′∗) ∼= Gal((K ′ ∩Kab)/K).

Proof. We have ArtK : K∗ ∼= W (Kab/K) and K ′∗ ∼= W (K ′ab/K ′) by Art′K


as well. Therefore, one hasK∗/NK′/K(K ′∗) ∼= W (Kab/K)/res(W (K ′ab/K ′)).Next observe that we have a surjection res : W (Kab/K) � Gal((K ′ ∩Kab)/K) with kernel res(W (K ′ab/K ′)). Indeed H = Gal(K ′ ∩ Kab/) isfinite and every σ ∈ H extends to an element in Gal(Kab/K) whose re-striction to Kur lies in < ϕK > an so we have surjectivity. Now anyσ ∈ res(W (K ′ab/K ′)) maps to identity in H since it fixes K ′, and anyτ ∈ W (Kab/K) that fixes K ′ ∩ Kab is a restriction of some element inW (K ′ab/K ′) so lies in res(W (K ′ab/K ′)). This ends the proof.

Putting together these results, we have

Theorem 5.4.9. ( Local Class Field Theory ) Let K be a local field.Then,

1. There is a unique homomorphism ArtK : K∗ → Gal(Kab/K) charac-terized by the following properties:• For a prime π ∈ K, then ArtK(π)|Kur = ϕK with ϕK the Frobeniusof Gal(Kur/K).• For an abelian extension K ′/K, we have ArtK(N(K ′/K))|K′ = 1.

2. For a finite abelian extension K ′/K, the Artin map induces the exactsequence:

1→ NK′/K(K ′∗)→ K∗ → Gal(K ′/K)→ 1.

Bibliography

[1] E. Artin, Algebraic Numbers and Algebraic Functions, AMS CHELSEAPUBLISHING 2005.

[2] N. Bourbaki. Commutative Algebra; ADDISON-WESLEY 1972.

[3] N. Bourbaki, Topological Vector Spaces, Chapter 1-5, Springer, BerlinHeidelberg New York 1987.

[4] C. Chevalley, On the theory of local rings, Ann. of Math., Vol. 44 (1943)pp. 690-708; available at: http://www.jstor.org.

[5] Henri Cohen; Xavier-Francois Roblot, Computing the Hilbert class fieldof real quadratic fields , available at: http://www.ams.org/mcom/2000-69-231/S0025-5718-99-01111-4/home.html.

[6] I. S. Cohen, On the structure and ideal theory of complete local rings,Trans. Amer. Math. Soc., Vol. 59, 1946, pp. 54-106; available at:http://www.jstor.org.

[7] P. Colmez Les Nombres p-adiques, Notes du Cours de M2 available at:http://people.math.jussieu.fr/∼colmez/.

[8] P. Colmez Corps Locaux, Notes du Cours de M2 available at:http://people.math.jussieu.fr/∼colmez/.

[9] A. J. Engler, A. Prestel, Valued Fields , Springer, Berlin Heidelberg2005.

[10] I. B. Fesenko, S. V. Vostokov, Local Fields and Their extensions, AMS,second edition 2002.

[11] A. Frohlich, Local fields in Algebraic Number theory, Academic Press1967.

[12] S R. Ghorpade, Balmohan V. Limaye, A Course in Calculus and RealAnalysis, Springer, New York 2006.

[13] M. Hazewinkel, Formal groups and applications, Acad. Press, 1978.

75

76 BIBLIOGRAPHY

[14] Franz-Viktor Kuhlmann, Valuation Theory, available at:http://math.usask.ca/∼fvk/Fvkbook.htm.

[15] Kenkichi. Iwasawa, Local Class Field Theory, Oxford University Press.New York 1986.

[16] J. Lubin and J. Tate, Formal complex multiplication in local fields,Ann. of Math. (2) 81 (1965), 380-387.

[17] J. S. Milne, Class Field Theory, Course notes available at:http://www.math.lsa.unich.edu/∼jmilne.

[18] J. Neukirch, Algebraic Number Theory, Springer, Berlin Heidelberg NewYork 1999.

[19] Xavier-Francois Roblot, Stark’s Conjectures and Hilbert’s Twelfth Prob-lem, available at: http://citeseer.ist.psu.edu/roblot99starks.html.

[20] Jean-Pierre Serre, Local Fields, Springer-Verlag, New York HeidelbergBerlin 1979.

[21] Jean-Pierre Serre, Local class field theory, Algebraic Number Theory (Cassels and Frohlich, eds), Academic Press, New York, 1967.

[22] Ehud de Shalit, Relative Lubin-Tate Groups, available athttp://www.jstor.org/stable/2045561

[23] J. Stevenhagen, Local fields, available at:http://websites.math.leidenuniv.nl/algebra/localfields.pdf.

[24] Paul B. Yale, Automorphisms of Complex Numbers, Mathemat-ics Magazine, Vol. 93, No. 3 (1966), pp. 135-141; available at:http://www.jstor.org.

[25] T. Yoshida, Local Class Field Theory via Lubin-Tate Theory, availableat: http://www.math.harvard.edu/∼yoshida/.

Index

i-th ramification subfield, 47p-adic absolute value, 13

absolute values, 11discrete valuation ring, 12system of representatives, 27upper numbering, 51

approximation theorem, 16Archimedean, 13Archimedean postulate, 13

complete, 22

formal AK-module, 60Frobenius kK-automorphism, 44

Hasse-Arf theorem, 69Herbrand’s theorem, 50higher ramification subgroups in lower

numbering, 46

incomplete, 22

local field, 36

non-archimedean, 13norm group, 52normalized, 13

p-adic valuation, 8prime, 26

ramification index, 38relative Lubin-Tate extension, 62residue degree, 38residue field, 17

tamelyramified, 40

Teichmuller representatives, 31the product formula, 21totally ramified, 39

ultrametric absolute values, 11ultrametric inequality, 13uniformizer, 26unit norm group, 53unramified, 39

valuation, 12valuation ring, 17value group, 11, 12

wildly ramified, 40

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Date post:	09-Feb-2022
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Local Class Field Theory via Lubin-Tate Theory

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