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Transcript
Bibliography*
Artin, E., Uber eine neue Art von L-Reihen, Abh. Math. Sem. Univ. Hamburg, 3 (1924)
89-108.
Artin, E., Beweis des allgemeinen Reciprozitatsgestzes, Abh. Math. Sem. Univ.
Hamburg, 5 (1927) 353-363.
Art in , E., Idealklassen in Oberkorpern und allgemeines Reciprozitatsgesetz, Abh.
Math. Sem. Univ. Hamburg, 7 (1930) 46-51.
Artin, E., Algebraic numbers and algebraic functions, Lecture Notes, New York Univ.,
1950-1.
Artin, E. and Tate, Jr., Class field theory, Lecture Notes, Princeton Univ., 1951-2.
Artin, E. and Whaples, G., Axiomatic characterization of fields by the product
formula for valuations, Bull. Amer. Math. Soc. 51 (1945) 469-492.
Barrucand, P. and Cohn, H., Remarks on principal factors in a relative cubic field,
J. Number Theory 3 (1971) 226-239.
Brauer, R., On Artin's L-series with general group characters, Ann. Math. 48 (1947)
502-514.
Brauer, R., Beziehung zwischen Klassenzahlen von Teilkorpern ein galois chen Korpers,
Math. Nach. 4 (1951) 158-174.
Bruckner, G., Charakterisierung der galoisschen Zahlkorper deren zerlegte Primzahlen
durch binare quadratische Formen gegeben sind, Math. Nach. 32 (1966) 317-326.
Cassels. J. W. S. and Frohlich, A., (ed.), Algebraic number theory, (Symposium),
Thompson Book Co., Wash. D.C., 1967.
Chat land , H. and Davenport, H., Euclid's algorithm in real quadratic fields,
Canadian J. Math. 2 (1950) 289-296.
Cheval ley , C., Class field theory, Lecture Notes, Nagoya Univ., 1953-4.
Chowla, S. and Selberg, A., On Epstein's zeta-function, J. Reine Angew. Math. 227
(1967) 86-110.
Cohn, H., Second course in number theory, John Wiley, New York, 1962.
Cohn, H., A numerical study of units in composite real quadratic and octic fields,
Computers in Number Theory, Acad. Press, London (1971) 153-165.
Cohn, H. and Cooke, G., Parametric form of an eight class field, Acta Arith. 30
(1976) 61-71.
*Bibliographies notable for completeness appear in Hilbert [1897], Hasse (Bericht) [1930b], Herbrand [1936], and Narkiewicz [1974].
267
Cooke, G., A weakening of the euclidean property for integral domains and applica
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273
Appendices
Olga Taussky:
E. Artin's Lectures on Class Field Theory (G6ttingen 1932)
Introduction into Connections between Algebraic Number Theory and Integral Matrices
275
Introduction to Appendices
ARTIN LECTURES
The three lectures by E. Artin were delivered in Gottingen in 1932. I was an
assistant at the Mathematical Institute whose main duty was to help with the editing
of Volume I of Hilbert's Collected Works. I was skilled at shorthand and had no
problem in taking notes. I wrote them up and the two other editors, H. Ulm and
W. Magnus, typed them on paper bought ourselves. In order to get reimbursed for the
paper, we sold them for one Mark apiece. H. Hasse checked the manuscript and sug-
gested more detailed proofs in some places. The treatment by Artin made use of the
ideas of Herbrand and was more modern than probably any other publication on some of
the basic facts in class field theory for some time. The notes were very much used
in spite of their limited circulation.
I am grateful to Mr. Robert Friedman for the translation.
MATRIX THEORY
Quite deep applications of matrix theory and operator concepts have been made
for some time on high levels of algebraic number theory. These will not be treated
here. Only connections with the basic concepts will be discussed. Complete proofs
have not always been included. The prerequisites of matrix theory are rather slight
and can be obtained from the now vast literature on the subject.
Algebraic number theory has been found most effective in giving structure and
deeper insight to many facts concerning rational integers. On the other hand,
integral matrices bring it back to the domain of rational integers enriching both
subjects at the same time.
Thanks are due to I. Reiner for criticisms concerning some of the details of
this appendix.
276
Olga Taussky California Institute
of Technology
March 1978
E. Artin's Lectures on Class Field Theory (Gottingen 1932)
Translated by Robert Friedman (Harvard University)
277
First Lecture
Class field theory deals with the decomposition laws of relatively abelian
fields, by establishing a connection between these laws and the division into
equivalence classes with respect to ideal groups in the base field. These divisions
into classes with respect to ideal groups are defined via congruences and are a
generalization of the usual division into residue classes in the rational number
field. The generalization occurs in the following way: First one defines a
division into residue classes for algebraic numbers and from this passes to a
division into residue classes for ideals. As the example of cyclotomic fields
already demonstrates, certain sign conditions are also necessary. Namely, for the
field of th m roots of unity, the decomposition law states that the primes p
which have the property p" 1 (m) split completely, 1. e. into prime ideals of the
first degree. Here one must, however, assume that p is taken to be positive.
Hasse remarked that this sign condition could be expressed as a congruence. This is
based on a convention which goes back to Hensel and which class field theory justi-
fies splendidly. Namely, symbolic "infinitely far" prime places are taken along with
the usual prime ideals of the field. They do not, however, appear as factors of an
ideal in the usual sense, but only as factors of the modulus with respect to which
the division into residue classes is being taken. Congruences with respect to them
mean only a sign condition. The relation a" 1 (p aJ) signifies that a is a
positive number. The statement that p is a positive prime number and has the
property p" 1 (m) is written in the form
For an arbitrary base field k, many sign conditions may occur. Namely, let
(1) (2) (r l ) k ,k , ... ,k be the real conjugate fields of k. Let a be a number
(1) (2) (r l ) and a ,a , .•. ,a the corresponding conjugates of a. We write the statement
that a(i) is a positive number similarly as a congruence, by assigning to each real
(1) (2) (r l ) field an infinite place j> 00 '1'00 , ... '1'00 and setting
278
a = 1 (.", (i») . - roo
It is expedient to define infinite places for the complex conjugate fields as well,
although congruence with respect to them no longer says anything. Otherwise the
infinite place would disappear in the transition from a real field to a complex
extension field. In order that a kind of decomposition law hold for infinite places (rl+l) (rl +2) (r l +r2)
as well, one introduces the infinite places roo 'roo , ... ,~ corre-
sponding to pairs of complex conjugate fields. We want to allow these infinite places
to appear in congruences as well, but these congruences are without content. This
corresponds to the notion of a totally positive number, which likewise imposes no
condition for a complex field. If K is an extension field of k, we call the
infinite places ~oo of K the factors, under an evident correspondence, of the
infinite places /"'00 of k. Here we make the convention that the J? divide the 00 1 !"Do once if they are real and twice if they are complex. If in addition one assigns
to each infinite place the degree 1, then one still has the theorem that the sum of
the degrees of all (equal or unequal) factors of a ~oo equals the degree of the
field extension. 2
We now form arbitrary moduli (Moduln) ""- from ordinary integral ideals and
infinite places. The symbol a=: 1 ('m.) will mean that the number a satisfies the
congruence a=: 1 with respect to the power of each individual prime ideal which
divides '111.. The symbol
a - S (rn,)
for a pair of non-zero numbers does not, however, means that a-S - 0 em) but rather
that
!! - 1 (.",) S
Following Hasse, the additive congruences which occasionally appear are denoted by
the symbol a =: S (~.) but here no infinite places are allowed to occur in ~. If
the numbers are prime to '111.., the additive congruence has the same significance as
l. This convention does not agree with that given by Hasse in his Bericht (Part la, p. 59).
2. If 1"00
is real, in general one must assume -f' divides roo once if both 00
are complex. 279
the multiplicative one.
After these considerations, we now divide the ideals into residue classes: We
define iT{:= 1( .... ) when 1) !Jl is a principal ideal (ex) and 2) the generator ex
may be chosen so that ex - 1 (m) holds. The relation I7t '= t (1<.) holds if f:-- '= 1 (:m).
There exist in general infinitely many residue classes with respect to a given
modulus, only finitely many if the modulus consists only of infinite places. If one
restricts oneself to residue classes prime to the modulus, then likewise only finitely
many occur.
The ideal groups form the basis for the division into ideal classes. The most
restrictive division into residue classes is given first via the relation of congru-
ence. However, one need not consider such a restrictive division into classes, even
for the decomposition laws of quadratic number fields. Instead one takes the union
of certain residue classes which form a group, to obtain an ideal group H, and the
union of correspondingly many residue classes to obtain cosets with respect to H,
in a manner which is easy to describe group-theoretically. Here, in fact, it is a
question of a system of complete residue classes modulo the discriminant D, namely
of the group H of numbers a for which (Q) = +1 while the only other class is a '
characterized by The decomposition law in a quadratic number field may be
easily expressed via this division into classes. In the general case as well, such
groups H, which consist of residue classes prime to a given modulus~, play an
important role. We wish in the general case as well to connect these ideal groups
and the corresponding division into classes with the decomposition laws of relatively
abelian fields. We thus arrive at the actual object of study of class field theory.
Let k be an arbitrary algebraic number field as base field, K a relatively
abelian extension field of k. One can find an ideal group H in k, defined
(erklart) by a suitable modulus ~v, so that the decomposition laws of K with
respect to k are regulated by the group H. More precisely states, this means:
All prime ideals in the same residue class modulo H undergo the same decomposition
in K. In particular, the ideal group H is so constructed that all the prime ideals
in it split completely.
280
The cyclotomic field of the full th
m roots of unity furnishes an example of
this. The prime numbers p which have the property p - 1 (mPa» split there into
prime ideals of the first degree only. Consequently H consists of all numbers a
for which a:: 1 (mp a> ) holds. If P doesn't lie in H, it lies in a residue
class with respect to H, and the prime numbers in the same residue class have
decomposition laws which depend only on the residue class. The general validity of
the analogous theorems for arbitrary relatively abelian fields is one of the main
results of class field theory.
In order to move toward a proof of these theorems, it is expedient to emphasize
the study of the norms of the field K relative to k. In fact, one arrives at
this by considering the following property of the decomposition in the cyclotomic
field: if P is a rational prime and f the smallest exponent such that f
p lies
in the group H, then f is the degree of the prime ideals into which p decomposes
in the cyclotomic field. Thus, the primes of the ideal group itself split into
ideals of the first degree, which states that the norm of such a prime factor is a
prime ideal in k. We must show the validity of this theorem in the general case
as well. Then clearly the following holds in particular: All the norms in k of
ideals of K lie in the group H.
In the case of a cyclic field, there exists a procedure, which in the final
analysis goes back to Gauss, to obtain more precise information about such groups.
One considers the groups which arise when one enlarges the norms NCR of ideals at
in K which are prime to a sufficiently high modulus ~ to full residue classes
mod~. In other words, one forms the product of the group of these norms N~
with the group of norm residues v prime to ~. We mean here by a norm residue a
number of the base field which is congruent mod ~ to the norm of a number A. If
v is prime to '1fl-, then it is always possible to choose the number A prime to
~ as well. We claim that this group H = NIl1' (v) is the ideal group under
consideration. To this end, we show that the number of residue classes into which
the ideals prime to the modulus split mod N~'(V) is at least as large as the
relative degree of Kover k. This demonstration is the goal of the first part of
281
the lectures and will be accomplished by purely arithmetic means. l,e will show in
the second part, using analytic means based on the methods of Dirichlet, that the
number of residue classes is at most the degree of the field extension. Both
statements taken together yield the proof that every cyclic field is a class field.
The decomposition laws are then easily ascertained with the help of the fact that
the degree of the field extension is identical to the number of co sets mod H.
We now prove that, for a suitable choice of the modulus~, the number of
cosets is at least as big as the degree n.
For the proof, Hasse has introduced relevant notation and developed the
group-theoretic machinery. One can establish a connection between the ideal groups
and other groups, about which one is able to say more, namely groups of numbers,
units, residue class groups, etc. Through isomorphic maps, one can gradually
reduce to such groups. On the other hand, one need not consider the basis represen-
tations of such groups. It is above all a question of the symbol for the index of a
group with respect to a subgroup. The group itself will be denoted by its charac-
teristic elements, e.g., an ideal group by the letter 61. The index which we have
to estimate can be written in the form (at: NIS1 (v», where at denotes the ideals
of k which are prime to the modulus 'm.. He claim:
(at: N(Jl(v» > n for a sufficiently high modulus ~.
First we rewrite this index by three simple group-theoretic rules:
1) If the three groups A,B,C are related by A2 B 2 C, then
(A: C) (A : B) (B : C) •
Applied to the index (at: NtSt (v», this yields, if we denote by a the numbers of
the base field prime to '17!.,
("l. : NIJ{ (v» (or: (v»
(Ncrt(v) : (v» «(I{: (a»(a) : (v»
(N at (v) : (v»
2) If AB denotes the product of the abelian groups A and B, [A,B] their
intersection, then
282
(AB : B) (A: [A,B]).
If we denote by 8 the group of ideals prime to ~ whose norm is a norm residue (the
principal genus), this rule applied to the index (J1: Nor· (\!)) yields
(Jt: (a))«a) : (\!)) (NOl.: N.~)
3) Let T(A) be a single-valued mapping of the group A with the property
that for every two elements A and A' the relation T(AA') = T(A)T(A') holds.
The mapping need not be one-to-one. We denote the solutions to the equation T(A) = 1
by AI' If we consider the index (A: B) and if the elements Al belong to the group
B, then (A: B) = (T(A) : T(B)). In general
(A: B)
if Bl stands for the solutions of T(B) = 1. Examples of such mappings T are the
passage from the group of numbers a to the group of principal ideals (a) or the
norm map.
We thus obtain, if we denote by E the units of the base field
(at: NOl (\!)) (Ul.: (a))(a : \!)
(E: [E,\!])(01:J:j)
We now apply Rule 1 again. We denote by a the generating substitution of the
cyclic extension and use the symbolic notation iJ{a instead of aOl. As the
group 13 contains the group OZl-a (A) letting A be the group of numbers of the
extension field prime to 'm. - we can on the basis of Rule 1 make the following
reformulation:
(il1.: N 01 (v))
1-0 hO (a : v) «(5 : <JL (A) )
I-a (E: [E,V])(:JZ::JZ (A))
Here we denote by hO the usual class number of the base field, which agrees with
the index (CR: (a)) of the groups at, (a) (which are prime to the modulus '/rl).
Next we consider the index (Ol: O1 l - a (A)), which we will show to be identical
283
to the number of invariant (ambigen) classes. l
As for exponentiation by 1-0, the following holds for ideals JZ as well as
for numbers A: The (l_o)th powers are identical to those numbers resp. ideals
which become 1 under the norm map. This is the famous Theorem 90 in Hilbert's
Zahlbericht: Every whole or fractional number A in K whose relative norm in k
is equal to 1 is the symbolic (l_O)th power of a certain integer B of the
field K.
In order to prove this, we will explicitly solve the equation 1-0
A= B • Here
the number B is of course uniquely determined only up to a factor from the base
field. If r is an arbitrary number, n the degree of the extension, then
has the property that
A 1-0
B , as long as B f O.
In order to prove that a B f 0 may be determined in this way, we let r run through
the powers of a generator 8 of the field. If B always yielded the value 0, the
determinant of the system of equations
o (i = 0,1, ... ,n-1)
would have to vanish. As this determinant is identical to a root of the discriminant
of the number 8 , it is thus by hypothesis different from zero.
It is likewise true for ideals that NCR 1 and en: = 8 1- 0 are equivalent;
in fact it turns out that
n-2 (3 1 + 01 + ()7l+O + ... + ml+o+ ... +0
where by the addition of ideals we mean module addition(formation of the greatest
common divisor).
1. I.e., those ideal classes which are invariant under o.
284
Using this lemma we now show that the index 1-0
(;JI. : JZ (A) ) is identical to the
number of invariant classes, i.e., the classes which remain invariant under the
application of 0. First remark that the restriction on Gl,A "prime to ...... " may
be left out without changing anything, as we are dealing only with classes of ideals.
This is important for the application of the lemma proved above, because the B
there, even for A prime to '7>1.., need not turn out to be prirr.e to '1>_'. According
to Rule 1
1-0 (01. : crt (A) )
( R: (A))
(<J!...1-0 (A) : (A))
But by Rule 3, if ~ runs through the ideals of the invariant ideal classes ~1-0
principal)
(at (A))
from this it follows that
(G1 : Q1.1-0 (A) ) cit: (A)).
We now rewrite the index (p: (A)). If we denote by e those numbers whose norms
are units, then J-0 _
()J- - (8). Then according to Rule 3, if we denote by ~ (the
invariant (ambige) ideals)l (resp. by (6)) those elements of ~(resp. of (A)) which
become one after exponentiation by 1-0,
(Ii: (A)) 1-0 t'¥
«8) (A ) ) ( .H 1 : (L'l)).
If E are the units of the extension field, then it follows, again by Rule 3, that
,,y 1-0 A./ (.;; : (A)) = (8: A E)(N 1 : (6)),
and as the principal ideals (a) of the base field are surely a subgroup of (L'l),
by Rule 1
1. Hasse has in the meantime found a new and shorter proof for the theorem,
(<rr: N m: (v)) ~ n, for which the hypothesis that the ideals at are prime to -m..
is removed. The considerations here about the group ~ become completely
unnecessary; instead, essentially the only invariant ideals are those of the base
field. 285
If we insert the ideals ~ of the base field into this, according to Rule 1, we
obtain, when we in addition apply Rule 3 in the denominator:
(ff: (A»
I-a I' (8: A E)(""~l: '>l. )(U( : (a»
(to : aE)
We have (01.: (a» ; hO• In order to determine (j} 1 : ·Tt), we reflect that every
prime ideal It" of the base field splits in K into
I-a As .41 ; 1, ~l must be made up of prime ideals of the base field and symmetric
product of conjugate invariant prime ideals of K, i.e., those for which e(1~ > 1.
Aside from ideals of the base field, we thus have ~; ~ f')r, r; 0,1, ... ,eC,)-l
and
(frl : 01.) ; n eCj» (,. finite)
Now, the only transformation of (%: (A»
(8 : A1-OE)
(to : aE)
which remains consists in applying the
norm map to the numerator of and exponentiation by 1-0 to the denomina-
tor. We obtain, by Rule 3, if we denote by n the units of the base field which
are norms of numbers of the extension field and by H the units of the extension
field which are the th
(1-0) powers of numbers,
(jJ-': (A»
(n : NE) n e <:;t-) " h (p.- finite) 0
If we combine this final result with the last obtained expression for (crc N<ll" (v»
(see above), we obtain
('J\. : N (l{ (v» CH : El-O) (a : v) (tJ : (Ttl-a CA»
(11 : NE) n e~1l-) (E : [E ,v]) 7' finite)
After yet another application of Rule 1 to (n: NE) we obtain
286
(Jt : Nell,.· (v» 1-0 1-0
(H : E )(a. : v) (tJ ::JZ (A» (S : T) (E : NE) II e(f') (E : [E, v])
(1' finite)
which finally leads, again by Rule 1, to the representation
(:n:NOl(v» (H : El - o)
(E : NE) (a : v) 1-0 II eC!) (/i:n (A»( [€,v] : n) .
(1' finite) >
In the sequel we will prove the following two theorems:
I. (H: El - O) n
(E : NE) II e'roo)
II. (a : v) II e (r) (1")
From these two theorems we obtain
(Ji: Nat· (v»
(The theorem of the unit principal genus).
for a sufficiently high modulus.
1-0 n([E,v]:T)(8:ff( (A»,
hence (01.: Not (v» > n for sufficiently high '"m.. The demonstration that
( at. : N 01 • (v» = n shows thus in addition that both other terms have the value 1,
Le., that for sufficiently high '1"- we have moreover:
1) ([E,V]: T) = 1 which states that every unit which is a norm residue
mod '?II. is the norm of a number.
2) The generalization of the theorem of Gauss that every class of the principal
genus is the th (1-0) power of an Ideal class:
(8: 1Jt1-0(A» 1.
287
Second Lecture
We now complete the proof that
(vt.:NJ\.·(v)) > n
by proving the two theorems I and II. Herbrand discovered that a certain group-
theoretic statement is the common source of both theorems.
Let Tl and T2 be two endomorphisms of a group A which mutually annihilate
each other, i. e. , Tl (T2(A) ~ TZ(TI (A») ~ 1. Let Al be the solutions of Tl (AI)
AZ those of T2 (A2) ~ 1- If B is a subgroup of A of finite index, which is
mapped by Tl and TZ to subgroups of B then, if Bl (respectively B2) are
defined in B analogously to Al (respectively A2) in A, the following holds:
(AI: T2 (A»
(A2 : Tl (A»
(Bl : T2 (B))
(BZ : Tl (B»
under the hypothesis, that the indices which appear on the left in the numerator and
denominator are finite.
Proof. First, by hypothesis, we actually have T2(A) <; AI' Tl (A) S A2. As
(A: B) is finite, by Rule 1 and Rule 3
(A: B)
(AI: TZ(B» (Tl (A) : Tl (B» (Bl : TZ(B»
(AI: TZ(A»)(T2(A) : T2(B»
(Tl (A) : Tl (B» (Bl : TZ (B»
The hypotheses are symmetric in the indices 1 and Z. Accordingly, if we compare
with that obtained when we first apply the mapping TZ' we obtain that this result (AI: T2 (A»
(Bl : TZ(B» is invariant after permuting the indices. This proves the theorem. 0
(H: E1- CJ) _ n Proof of Theorem I: -=---;---:-
(E:NE) - TIe(JVrn )' if (as above) H denotes the
units of the extension field which are (l-CJ)-th powers of numbers.
Herbrand has shown, with the help of representation-theoretic methods, that
288
1,
the Minkowski unit theorem can be generalized to relatively Galois fields. A proof
of this theorem which makes no use of representation theory can also be given (see
Artin, Henselfestschrift l ). This generalized Minkowski theorem implies in particular
for the cyclic case: There exist r+l units Hl,HZ, ... ,Hr +l which, along with
their conjugates and a system of independent units of the base field El,EZ, ... ,Er ,
generate a subgroup E of finite index in E. The only relations which hold among
these units are the following:
ni-l H~+a+ . . . +a 1, ~
where n. ~
(i = 1,Z, ... ,r+l).
By the group-theoretic lemma of Herbrand, we need only carry out the calculation of
the index quotient for the subgroup E. The elements which are mapped to one by the
norm map are the Hi and their conjugates. On the other hand, no unit of the base
field is mapped to one, because the possible roots of unity in k do not appear in
E, by definition. The elements which are mapped to one by I-a are the units from xl x2 xr
El E2 ···Er the base field We replace H by Fl (a) F2(a) Fr+l(a)
H = HI H2 ···Hr +l where
the Fi(a) are polynomials in a with integral coefficients. If we expand these
polynomials with respect to I-a, we see that H can be put in the form
H
Here the xi run through the interval for it follows from the identity
nl-l 1 + x + ... + x - n i mod(l-x) that, as a consequence of the relation
n -1 1+a+ ... +o i
Hi
among the Hi'
El - a by iI-a
1,
the
and
n. H ~ lies in ~
representation
thus obtain in
the group iI-a
is unique up to
the numerator
The denominator is replaced by the index
1. See J. Reine ~Angew. Math. 167 [1932] 153-156.
II n. ~
As no other relations hold
powers.
r+l n
r n •
We replace
289
The quotient is thus
(H : El - o) (E: : NE)
n
n e ( (i)) roo
Proof of Theorem II. (a: v) n e(l') for a sufficiently high modulus ?It..
~)
For the proof of this theorem, the modulus ""1'IV must be chosen sufficiently high.
We construct ~ as a product of divisors of the discriminant to sufficiently high
powers and of the infinite ramified places. If
the norm residues with respect to the modulus
'11t. = n f' A and we denote by
~ A which are prime to '7}L., "
(a: v) = n(a: v A)'
r This follows from the fact that the relations
a - NA(1t\.) and
A are equivalent, if 'fIL is the product of the ;;Vi For if we write '~ as the product
of two relatively prime ideals,
determine a number A such that
"\ and ""'2 and a::: NAl (")), a ::: NA2 (7lLz) , we can
A::: Al("li), A::: A2 (""'2)' Then a::: NA("'l) and
We thus need only show that, for sufficiently large A, the following relation
holds:
e Cf) .
For f =;too the congruence
is always satisfied unless the field k is real for roo and the extension field K
is complex for JVoo ' In this case, the congruence is satisfied if and only if a
is positive. The index is then 2, and that is precisely the value of e(~). The
real difficulty lies in the finite primes. To deal with them, we will usejc-adic
methods.
Let P be a prime factor of 1" in K. The statement that
290
is equivalent to
where Z is the decomposition field of fJ and t denotes the unique prime ideal
of Z which is divisible by 1'. In fact, if 11 is the Galois group of Kover
k, }- the decomposition group and
the decomposition of ~' into the cosets of "
I' then
1+°1+" .+0 -1 ~'. (A n ),J
but this is the norm of the number B 1+°1+" '+On-l
A relative to the decomposi-
tion field. The converse if proven in the following way: If a = NK/zB~r), we
solve the simultaneous congruences
-1 r A = 1 (Ol~! )
!)
-1 r A = 1 (am_I'}')'
I
r This system has a solution, because ~ is different from the
the decomposition group. If we then apply the substitutions
these congruences, we obtain
r A = B (g- ) °1 r
A - 1 (t)
°m_l r A _1('1-')'
-1 r °i f' as
Hence Hal + •• '+Om_l
A _ B (tr ). If we now exponentiate by ?' we obtain
to
291
Thus
and as this is a congruence between numbers of the base field, it also holds r mod,f •
As every number of the decomposition field is congruent to a number of the base
field mod an arbitrarily high power of the prime ideal, the residue classes are
completely represented by numbers of the base field. For the calculation of
(a : v A)' t
we may thus restrict ourselves to the special case where the base field
is the decomposition field. He now show that the statement A a =: NA0r) , for
sufficiently large is equivalent to a being a f-adic norm. For if 0.- 10/),
then the ,f-adic logarithm log a surely exists. If now A is large enough, log a
also has arbitrarily high order in t. Hence we can form the 1-adic e-function of
1 ; log a, and we have
a
th so that a is not only a norm but even an n power. If a is an arbitrary norm
residue, then a - NA<f)' hence ~A == 1 (th, thus JL NA is a radic norm NC.
Hence we have a = N(AC).
He can thus calculate the index (a: NA) instead of (a: v), where now a and
A are numbers, in the f-adic fields and K j- associated to k and K, which
are prime to r. "Theorem 90" in Hilbert's Zahlbericht holds for t-adic fields,
i.e., if NA = 1 f-adically, then A . th 1S a (I-a) power. He denote by Al
group of numbers of K f·
prime to if' whose norm is ;-adically 1 and form the
quotient
(AI: AI-a)
(a : NA)
the
This index quotient may be calculated by the group-theoretic lemma of Herbrand, by
passing to a subgroup of finite index. It will turn out that this quotient has the
value 1, so that
(a : NA)
292
The calculation of (a: NA) is thus reduced to that of 1-0
(AI : A ) , which may be
not dealt with in the following way: VIe denote by B arbitrary numbers of Kf '
necessarily prime to 1" by 13 arbitrary numbers of kt , and get from Hilbert's
"Theorem 90" and Rule 3 that
(B : SA).
As the numbers of the base field, if they are divisible by 1', must be divisible by
pe<t'), we have (B: SA) and hence also
(a : NA)
All that remains is to check that the above mentioned index quotient
actually has the value 1.
(AI: Al - o)
(a : NA)
To calculate the quotient, we choose a suitable subgroup of A of finite index.
First consider the subgroup of A formed by those numbers which are = 1 mod a
sufficiently high power ;Y We can then pass to the logarithm, as this is a
one-to-one mapping. The numbers which are = 1 mod ~~ are mapped into a group of
numbers B = 0 mod jY. We denote by S(B) the trace of B and by BO the group
of numbers whose trace is O. The entire index quotient is transformed by this
mapping into
(BO : (l-o)B)
(13 : S(B»
here i3 is again the subgroup of B ~lhich is invariant under o. As r is a
principal ideal in the f-adic field k, we can divide by ~ this is again an
isomorphic mapping. Thus the condition B = O(~) is completely unnecessary. Now
we fix the subgroup of finite index which will enable us to calculate the quotient
once and for all. We let B run through only those numbers
B
293
where C denotes an integer of the field K which, along with its conjugates,
generates a basis of the field, and the ai are arbitrary integers of the base field.
Then
S(B) as(e)
where a is an arbitrary integer of k. Hence the numbers B are identical to the
numbers S(B). It follows similarly that the numbers BO are identical to (l-a)B,
so that both numerator and denominator and hence the entire quotient
(Al : Al - a )
(a : NA)
This completes the proof of the relation
1.
(ot: N r1l (v)) > n
for sufficiently high ~~. It is in fact enough to consider an <~ which is composed
only of primes of k which ramify in K to sufficiently high powers.
He now prove that
(oc: N <R (v)) < n,
by further developing the methods of Dirichlet. He introduce the L-series as a
technical device. Let H be any ideal group with respect to (erklart nach) a
modulus ~. We form the division into cosets associated to H. Let h be the
ideal class number. Let X be the h characters of the abelian ideal class group
with respect to H. The L-series associated to this ideal class group look like
1 e L ~.
(j5) rN~;s (1,'''-) =1 '
L(s, X) II xr-..) (1. -m) =1 1 -~ '.r' s Nr
One shows, via the elementary theory of Dirichlet series and a rather complicated
determination of the density of ideals ordered according to increasing norm: The
given representations are absolutely convergent for s > 1. At the point s = 1, the
L-series of the principal character has a pole of order 1, and the L-series of char-
acters distinct from the principal character are regular there. We will show that
they do not vanish.
294
Let K be an extension field of k, where we assume that the ideal norms of
all ideals prime to '"111- lie in H. We have
In the sum in the exponential, the contribution from all prime ideals of degree
bigger than one is convergent for s = 1. Hence, if we denote by m<t,) the number
of prime factors of degree one which r has in K,
Analogously, we can write the product of the L-series over all h characters as:
TIL(s,X) X
= e
1 h L -- + 1/1(s) s r6H Nkt'
Here <I>(s) ,1/1(s) are functions of s which remain finite for s + 1.
If we now apply to K the hypothesis that the ideal norms of all ideals prime
to '1IL lie in H, then, up to finitely many prime ideals, the only terms which
remain in the sum are ideals of H. As has a pole for s = 1,
there must be an infinite number of these. But it follows from this that the L-series
cannot vanish at s= 1. For if even one vanished, the pole would be cancelled out in
the product of all L-series, and the right-hand side would be regular. But this
contradicts the fact that the sum in the exponential diverges. Thus, none of the
L-series can vanish for s= 1. Because of this, Dirichlet's proof on arithmetic
progressions applies. By using the well-known character formulas, one can easily
conclude that prime ideals lie in every residue class mod H with density I/h:
If rv runs through the prime ideals in H or in the same coset mod H, then we
always have
1 1 h log s-l + <l>l(s) ,
when s+ 1.
295
In particular, let :tv run through the prime ideals in the ideal group
us compare this result with the representation of sK(s):
!EJll n
1 1 1 n log s-l + ¢2(s)
H. Let
where for s ~ 1. mCj) is the number of prime factors of l' of
degree one, hence m<1) .::. n. Thus
n < L
mCj» >0
~_l_ <
the last inequality because, by the hypothesis on K, the ~ which appear in the sum
belong to H. The finitely many .1> 1""'- may be ignored, their contribution may be
absorbed into ¢2(s).
Thus, the same inequality must also hold for the right-hand sides, from which it
follows that
1 - > h
I
n' i. e. , n> h.
Now, we showed earlier that, in the case of a cyclic field and for H = N ~ (v),
~~ sufficiently high, then h = ('I.: N a1.. (v)) ~ n. On the other hand, H is a
group to which the hypotheses made above apply, as it contains all norms of ideals
of K prime to 71L. In these circumstances, we have just shmm that n> h.
Together these prove, for the case of a cyclic field K, the relation
n = h
for a certain group H.
If H is an ideal group mod 1~ in k of index h, K an extension field of
relative degree n, if all norms with respect to k of ideals in K prime to om
lie in H and if n = h, then we call K a class field over k. This is Takagi's
definition with a slight modification. We have thus shown that every cyclic extension
field is a class field with respect to a suitable group.
296
In order to develop the theory further, one can replace this definition of a
class field by another, more analytic definition, which is more suitable for many
purposes. In the sequel, when we wish to prove that a field is a class field, we
will sometimes use one definition and sometimes the other. Namely, we replace the
second condition n = h by the requirement: Almost a1l prime ideals of the ideal
~ H split completely in K, i.e., into distinct prime ideals of degree one.
Here "almost all" means that
~l
log s-l o
where we extend the summation over all those prime ideals which fail to satisfy the
condition. These two definitions of a class field are equivalent. If namely n=h,
then (see above) m(f) must equal n in "almost all" cases. If conversely m(,r) = n
in almos~ a1l cases, then n = h. (While it is sufficient to assume for the definition
of a class field that "almost all" prime ideals split completely, we shall prove that
all prime ideals of the ideal group split completely.)
297
Third Lecture
With the help of the second definition of a class field, we will prove the
following composition theorem for class fields: If two fields Kl ,K2 are class
fields with respect to the ideal groups Hl and H2, then their compositum K1K2
is a class field with respect to the intersection [Hl,H21.
Proof: If we factor the norms of ideals of the compositum through
through K2 , we see that they must all lie in the intersection of the two ideal
groups. Next, almost all prime ideals of [Hl ,H21 split completely in K1K2. For
almost all prime ideals of the intersection split completely in both Kl and K2.
But if a prime ideal Jv of k splits in both Kl and K2 into different prime
ideals of degree one, then ~ likewise splits into different primes of degree one
in K1K2. In fact, consider the smallest Galois extension field K of k which
contains Kl and K2. Let i J be a prime factor of ;- in K. The decomposition
field of f contains in particular every subfield of K for which l' is unramified
and such that the prime ideal belowP has relative degree one. The decomposition
field thus contains Kl and KZ and their conjugate fields, as ~ splits in all
of these into prime ideals of degree one. It is therefore identical to K. Thus
splits in K and consequently also in K1K2 into distinct prime ideals of the first
degree only.
As every cyclic field is a class field and every abelian extension is the
compositum of cyclic extensions, the composition theorem implies the Takagi converse
theorem: Every abelian extension is a class field.
The ordering theorem: If the class field K2 contains the class field Kl ,
then the ideal group H2 associated to K2 is a subgroup of the ideal group Hl
of Kl , and conversely.
In fact, K1K2 = K2 is a class field with respect to the intersection [Hl,HZ1.
As this intersection is identical to HZ' HZ is a subgroup of Hl . We prove the
converse as follows:
It is a class field with respect to H2, hence the degree of K1K2 is the same as
Z98
that of K2. As KlK2 contains K2 , they are identical.
A special case of the ordering theorem is the uniqueness theorem of class field
theory. For every ideal group H in k, there exists at most one class field K.
From this it follows immediately that every class field is relatively Galois
over the base field. For every conjugate field K' is, like K, a class field.
Now that we have proved the composition and ordering theorems, we turn to the
translation theorem of Hasse: If K is a class field over k with respect to the
ideal group H, and if ~ is an extension field of k, then the compositum ~K
is a class field over ~ with respect to the group H of those ideals of ~ whose
norm lies in H. (A. Scholz later found a proof of this theorem which may be given
at the beginning of the theory).
We first check that H is defined (erklart) with respect to the same modulus
.~ as H in k. It is enough to show that every principal ideal (A) with the
property (A):: I ('7/~) lies in H. In fact, the conjugates of (A) are then also
I (7/t,) , as '11'- is an ideal of the base field. Thus the product (NA) is also
I em.). This is still true if '11l. contains infinite places, one need only remark
that the norm from a complex field to a real field is always positive. As (NA)
lies in H, (A) is thus contained in H. We now show that ~K is a class field
over K, and prove first that the norms of ideals prime to ~ lie in H. We
accordingly factor the norms relative to k through K and through ~. If we
factor them through K, we see that they lie in H. Hence the norms relative to
~ lie in the group H. For the second criterion, we use the property that almost
all ideals of the associated ideal group split completely in the class field. We
thus want to show that almost all prime ideals ?- of H split completely in ~K.
Here we may restrict ourselves to prime ideals of absolute degree one. The relative
norm of t is then a prime ideal l' in k which belongs to the group H. We have
Nt = N~ and thus an exceptional set of Dirichlet density 0 of the r s corre
sponds to a similar one for the 1's. (We assume absolute degree one only because
of the application of the Dirichlet series). Almost all }V split completely in K.
Hence we may conclude as above that almost all l' split completely in ~K.
299
It follows from the translation theorem that: Every field lying between the base
field and a class field is a class field (Hasse). In fact, if we choose as our field
~ a subfield of K, it follows that K is a class field over ~ with respect to
the group H of all ideals whose k-norm lies in H. The norms of ideals prime to
the modulus ""- lie in a group H' mod 71l- which contains H. We show that the index
of this group is equal to the relative degree n' of Dover k. By the trans la-
tion theorem, the index of the group H in ~ is equal to the degree of the field
K over ~ , namely n/n', which follows from the multiplication law for field
degrees. If we form the norms of D-ideal classes mod ~ relative to k, this is
an isomorphic mapping under which the group H corresponds exactly to the group H.
The index of H in H' is thus likewise equal to n/n', and the index of H' is
thus n'. As on the other hand H' by construction contains the norms of ideals of
~ prime to ~, it follows from the first definition of a class field that ~ is
a class field.
As every subextension is a class field and every class field is Galois, we can
already say something about the structure of the group of a class field. Namely,
every subgroup of such a group is a normal subgroup. Hence the group itself is a
Hamiltonian group. We will actually prove that the group is abelian; this will
follow from the decomposition law. The demonstration of the decomposition law is
based on Hilbert's theory of the decomposition group. We will determine that not
only do "almost all" prime ideals of the ideal group split completely, in fact every
prime ideal in the principal class splits completely in the class field into primes
of the first degree. We will prove generally the decomposition law for unramified
If f is the smallest positive number such that f r lies in the
principal class H, then ~ splits in the class field into distinct prime ideals
of relative degree f.
The decomposition group of the unramified primes is cyclic. If f is the
decomposition group of a prime factor P of ?' and if Tf' is a conjugate prime
ideal, then, as is well-known, T-ltT is the decomposition group of Tf'. As every
subgroup of the Galois group of a class field is normal, the decomposition groups of
300
all prime factors of a prime ideal Jv are identical. We may thus speak of the
decomposition group of 1" itself.
We will deduce the decomposition law from the fact that the decomposition groups
of the unramified primes are cyclic. First we prove: Every prime ideal ~ in the
principal class splits completely in the class field. At present we know only that
almost all prime ideals in the principal class split completely and that a prime
ideal which does split completely lies in the principal class.
Let then K be of degree n over k, ~ a prime ideal of H. We form the
compositum of K with another field K* disjoint from K, whose degree n* over
k is divisible by n and in which ~ is undecomposed. We assume also that K*
is a class field over k with respect to the group H* and finally that n* is
n* r lies in H*. We will later show how we can the smallest number such that
relax the hypothesis that 1" is undecomposed in K*. This is important, because the
field K* which satisfies the above-mentioned conditions will be some cyclotomic
extension of k, and one can show that the other conditions will be satisfied only
if one replaces the indecomposability of ;V by a weaker hypothesis.
We will conclude that tv' splits completely in K by showing K is the
decomposition field of KK*. In a Galois decomposition field, however, a prime
ideal always splits into degree one primes only. Hence we need only show that, with
our hypotheses on K*, the field K is the decomposition field of KK*. Let K
have Galois group ,~ over k, K* group ~*. As we also know that the elements
of K are unchanged by the application of .q* and the elements of K* by an
application of .~, the Galois group of KK* is the direct product ~Jr*. By the
composition theorem, KK* is again a class field over k. The decomposition group
r of ~ in KK* must be cyclic and must have order at least n* , as f is
undecomposed in K* and thus has degree at least n* in KK*. The order of the
decomposition group must thus be a multiple of n*. As the group is cyclic and nln*,
the order must equal n*. Hence the decomposition field KZ has the same degree n
over the base field as K. As a subfield of the class field KK*, KZ is likewise
a class field with respect to a group Hz which contains the intersection [H,H*] =~
301
it must lie in HZ thus contains the group As ;f splits completely in
,'I" "- 2/r n *-1,<1-;;+II.+/, +···+f· A'. This group is contained in H. As there exist n classes
mod H, nn* classes mod l' and the above classes by the definition of n* are
distinct, this group is identical to H. Thus KZ is a subfield of K, and as
they have the same degree over k, we can conclude that KZ is identical to K.
We need only show how we may relax the hypothesis that ~. is undecomposed in
K*. Namely, it is enough to arrange that f splits in K* into.possibly several
prime factors of degree n* divisible by n. We then change the base field and
pass to the decomposition field K* Z
of t in K*. We now reach our goal with the
help of the translation theorem: 1" has the prime factor t' of degree one in K~.
This prime factor stays undecomposed in K*. In KK~/K~, on the other hand, ~
splits completely. Let H correspond under the translation of base field to HI'
If f lies in H, then r- lies in HI' The hypothesis for the base field (here
KZ) that there exist a suitable field K* of degree n* divisible by n in which
t' is undecomposed is here applicable. Thus t splits completely in KK~, by the
proof above. But then jv must also split completely in K.
As for the construction of the auxiliary cyclotomic field K*, we will only
say that it is based on the following lemma about rational numbers: Given a prime
p and a natural number n, one can find infinitely many relatively prime m so
that the order of p mod m is divisible by n. This can be reduced to: In the
field k n of th n roots of unity, there exist infinitely many primes of the first
degree which are undecomposed in
We have thus shown that every prime ideal in the principal class splits
completely and prove now the general decomposition law for unramified primes: If f
is the smallest number such that ,! lies in the principal class, then ~ splits
in the class field into distinct prime ideals of relative degree f.
In contrast to Takagi, we prove this theorem with the help of the translation
theorem.
If the degree of a prime factor P of 1" in the extension field were f''; f,
we form f'
NP =./'- As P N belongs to the principal class and f was the order of
302
fv relative to H, f divides f'. Let KZ be the decomposition field of ;V. Jl
splits completely in KZ and the degree of the whole field with respect to the
decomposition field is f'. K is a class field over Kz with respect to the ideal
group H of all ideals whose norms lie in H. Let t be a prime factor of I'- in
tv= Nf the
fth power of
fth power of
f
r is the smallest such that ,t is also the smallest such that
lies in H and as
f
t lies in H. We
have f < f' and fl f'. The whole field is cyclic over Kz of degree f', hence
there must exist a subextension KI of KZ of degree f. Let be the ideal
group in KI which is determined by K. The prime ideal t of KZ remains
undecomposed in K, hence does not split in KI either. In order to determine the
situation of r relative to one must consider relative to We have
As however at lies in H, ~ also lies in the group HI' In the sub-
extension KI , i' thus belongs to the principal class and hence splits in the class
field K. As on the other hand t stayed undecomposed, we have reached a contradic-
tion to the assumption f' f f. With this we have proved the full decomposition law.
A consequence of the decomposition law is the isomorphism theorem: The Galois
group of the class field is isomorphic to the class group with respect to H.
We first prove this theorem in the cyclic case: the class group with respect
to H is cyclic if and only if K/k is cyclic.
If K/k is cyclic, then there exists a prime ideal f' which is undecomposed
in K. If the degree of the extension is n, there thus exists a ;f for which the
th n n power is the smallest such that ;; lies in the group H. Hence the class
group with respect to H is cyclic.
Conversely, if the class group with respect to H is cyclic and a class field
exists for H, the theorem on arithmetic progressions holds. Thus there exists a
prime ideal in the generating class of order n, n so that f ~ 1 for n and for no
smaller natural number. This prime ideal must thus be undecomposed. But this is
only possible in the cyclic case.
In order to prove the general isomorphism theorem, we first show the following
deep theorem: if there exists a class field K for H and if HI J H, then there
303
exists a class field which is contained in K. We choose a prime ideal j- of
the group Hl . Let f be the smallest power such that F € H. We then consider the
decomposition field KZ. It follows from the decomposition law that the corresponding
group HZ must contain the group f-l
H+fH+ ••• +f H. It follows from degree con-
siderations that HZ is identical to this group. We have thus shown that, if there
exists a class field for H, then there also exists a class field for HZ. We know
that there exist prime ideals in every ideal class. If now Hl is not exhausted by
Hand r, we can proceed in the same way until Hl is filled up. This proves
the theorem that, if there exists a class group for H, then there exists a class
field for every larger group.
It is now easy to prove the general isomorphism theorem: I,e think of the ideal
classes as represented by a basis Cl ,C2, ... ,Cs ' where the orders of the basis
classes are f l ,f2 , ... ,fs . We construct the s ideal groups
The class group with respect to H. 1
is cyclic, hence the class field associated
to Hi is cyclic over k, as we have already proved above. The compositum
KlK2000Ks is a class field with respect to the intersection
hence it is the given field K. As the composition of cyclic fields, it is abelian,
and the Galois group is plainly isomorphic to the class group with respect to Ho
304
Introduction into Connections Between Algebraic Number Theory and Integral Matrices
305
A matrix with elements from a ring R will be called an R-matrix. By integral
matrix is meant a Z-matrix.
Some of the links between algebraic number theory and integral matrices are
quite deep, but only a few basic relations will be mentioned. Some of the more
advanced results are indicated in square brackets.
By isomorphism the ring of algebraic integers in an algebraic number field can
be treated via matrix theory. Studying the ring generated by a general integral
matrix leads to a generalization of algebraic number theory.
The links between the subjects enable algebraic integers of degree n to be
immersed into the matrix ring Znxn which is a noncommutative ring. This adds new
problems and new methods to both subjects. Alternatively, certain results can be
generalized by viewing numbers as lxl matrices and lifting them up to statements
about the nxn case. Also, some results concerning determinants can be better
understood through their matrices.
For a long time there had been a great many interactions, to and from, between
algebraic number theory and noncommutative concepts. The most famous interaction is
the use of Galois Theory, a more modern idea is the use of the integral group ring
of the Galois group. On the other hand, algebraic number theory stimulates number
theory in the subrings of noncommutative algebras to which the group algebras belong.
Integral matrices are noncommutative and this fact helps to shed much light on
commutative number theory.
The following connections are basic:
(a) The characteristic roots of an integral matrix are algebraic integers.
(b) Conversely, consider an algebraic integer a. Let it be a zero of an
irreducible monic Z-polynomial (i.e., a polynomial with coefficients in Z) f(x)
n n-l x +alx + ••• + an_lx+ an of degree n. Let A be an nxn Z-matrix with charac-
teristic polynomial f(x). Then the rings Z[aJ and Z[AJ are isomorphic. There
always exists such a matrix, e.g. the left companion matrix of f(x):
306
o 1
The matrix A has many features which are connected with a, but not displayed
by it directly. The reason is, that while a is unique apart from algebraic conju
gation, there is an infinity of suitable A's. This will be one of the major issues
discussed in some detail here, in connection with ideals and ideal classes.
(c) Of particular interest are the unimodular Z-matrices i.e., the square
Z-matrices of determinant ±l. Their characteristic roots are units in algebraic
number fields. The units in a given algebraic number field are represented by
matrices in a subgroup of the unimodular group, also called GL(n,Z), while
SL(n,Z) refers to the case of det +1. However, the set of all unimodular matrices
is meaningful for every algebraic number field, in connection with change of bases
of modules. Similarly, in some way, the set of all algebraic units is meaningful
to a particular field.
(d) The discriminant of an algebraic number field leads to another connection.
By definition, this is an integer which is the determinant of a special integral
symmetric matrix. In recent years some emphasis has been given to the study of the
whole matrix, the discriminant matrix, and its corresponding quadratic form, the
trace form.
(e) Norms from algebraic number fields are closely linked to matrices, via
representation theory. In recent years a number of theorems have been observed
where norms from algebraic number fields turn up connected with noncommutativity of
integral or rational matrices.
From now on we study a fixed algebraic number field F of degree n with 0
the ring of integers in F, the so-called maximal order in F where order means
any subring 0 of 0 contained in F which satisfies:
1) 1 € 0,
307
2) 0 contains an independent Z-basis of n elements which implies Qo F.
We now devote four sections to our main goal.
1. Ideals and ideal classes in F
Let be a basis l for the order o. Then every ideal ilZ in o has
a Z-basis of n elements, say al, ... ,an . It is well known that such a basis exists
for every ideal. A routine method for obtaining one was devised by MacDuffee using
Z-matrices and their greatest common right divisors. Outside of the case of quadra-
tic fields, little attention has been given to this problem.
MacDuffee's method consists in representing the given field F, with
WI"",Wn as basis for its ring of integers, as an algebra of Q-matrices with a
basis of the corresponding nXn Z-matrices. Let M(a) be the representation of
a € F obtained via this fixed representation.
Theorem 1 (MacDuffee). Let the ideal ~ be given as the greatest common
divisor of the integers al, ... ,ak . Then a greatest common right divisor of
is a Z-matrix B -- --- with the property that B applied to the
basis gives a Z-basis of at.
A matrix like the above B is called an ideal matrix with respect to the basis
and the Z-basis of 0"1..
Definition (Poincare, MacDuffee, Taussky). The Z-matrix A is the ideal matrix
of the ideal crt, with respect to the above sets of bases if
Since all bases of
of ~ are of the form
the form UAV.
{') -C) n n
a are of the form vet) , v
{), U
n
w n
unimodular, all ideal
unimodular and all bases
matrices of (fr are of
1. The term basis will be used instead of Z-basis when no misunderstanding is likely to occur.
308
[An analogous concept cannot be immediately introduced in algebraic extension
fields of F because of the lack of an O-basis. However. several generalizations
have been studied. Of particular importance for this generalization is the work of
Steinitz who showed that the ring of integers in an extension of relative degree m
has m+l O-generators.]
It is known that I det AI norm ~ where norm of an ideal is the cardinal of
the residue class ring of o/~.
The Smith normal form S(A) = diag(sl ••••• sn) is of special significance. For
its existence implies that for a suitable basis of a and of ~ we have
i = l ..... n.
A more general statement obtained by Mann and K. Yamamota. with a special case
obtained by Taussky previously. is:
Theorem 2 (Mann-Yamamoto. and Taussky). Let ~ ••.•• ~ be a set of ideals in
the ring 0 of integers of an algebraic number field. Then there exists a basis
for 0 such that all these ideals have a basis of the form
i=l ..... n.
ExamEle. Consider the ring Z [i] and the two ideals :J{l = (1 + 2i) •
0{ = 2
(1 - 2i). The ideal 011 has the Z-basis 1 + 2i. -2 + i and 0{2 has the
(1 2 Then the desired basis for 0 is B (:) Z-basis 1- Zi. -2 - i. Let B = 3) . 1 1
1+ 2i (1+3i)' replacing the ideals by (1(1+2i}.5(1+3i}}.(5(1+Zi}.1(1+3i}).
Two important theorems concerning ideal matrices will now be mentioned.
Let wl •...• wn be a basis for 0 and let X be an ideal matrix for the ideal
with basis al •...• a n . Let
i = l ..... n
where Xi are Z-matrices. The matrices Xi span over Z a ring X isomorphic to
O. Then the following theorem holds.
309
Theorem 3 (MacDuffee, Taussky). The matrix X is an ideal matrix for 0 if
XXX-l is again a ring of integral matrices. (See also Theorems 6 and 7).
Theorem 4 (Taussky). Let X~, be ideal matrices for the ideals in
O. Then there exists a unimodular integral U such that A U B is an ideal matrix
for iJtA. [The above theorems have been extended to relative fields by G.B. Wagner
and S.K. Bhandari and V.C. Nanda.l
Ideal matrices will now be connected with ideal classes via classes of matrices.
Definition. Let A be a Z-matrix. The set of matrices s-lAS. with S a
unimodular Z-matrix, is called a class of matrices.
This is clearly an equivalence relation.
Theorem 5 (Latimer and MacDuffee. Taussky). Let f(x) be an irreducible (monic)
Z-polynomial of degree n. Let a be a zero of f(x). Then the number of matrix
classes generated by nXn Z-matrices A for which f (A) = 0 is equal to the number
of ideal classes in the order Z[al and therefore finite.
Proof. Let A be an nxn Z-matrix with f (A) = O. Then A has a as charac-
teristic root and f(x) as characteristic polynomial. The (unique) characteristic
vector of A corresponding to a can be chosen to lie in Z[al. Let such a vector
have components al •...• a n . Then
{:') -{') n n
This equation implies that a Z-combination of the a j is equal to aa i for
i = 1, ..• ,n. Similarly, p (a)a i , i = 1, ... , n, if a Z-combination of the a j for any
Z-polynomial p(x) via peA):
This shows that the a i form a Z-basis for an ideal in Z[al. This ideal is unique
apart from a common factor of the a i . Hence, only the ideal class of the ideal
310
corresponds to A. If instead of A another matrix s-lAS in its class had been
considered, then 5-1('}l') would euro up .0 eh.c.,e,ci,'i, ."eoc. Sine, 5 i,
n unimodular, this gives the same ideal referred to a different basis.
Conversely, let a Z-basis Bl, ... ,Bn of an ideal be given. Then aBi n I b· k8k , i = 1, ... ,n, with bk € Z. Then B = (bik) has the property that
k=l ~
61"" ,Bn is a characteristic vector of B with respect to the characteristic
root a. If instead of B1"",Sn another basis T(Sl"",Sn) with T unimodular,
had been considered, then B would be replaced by TBT-l .
The above facts establish a 1-1 correspondence between the matrix classes
S-lAS, for all A with f (A) = 0 and the ideal classes in the order Z [a].
Latimer and MacDuffee obtained a corresponding theorem for Z-polynomials not
necessarily irreducible over Q.
Examples (Latimer and MacDuffee);
1. Let f(x) be x4+x3+ x2+x+1. Then a is a 5th root of 1. It is known that
the field generated by this number a has 2 3 l,a,a,a as a basis for its ring of
integers, so that Z[a] is this ring. The field generated by this a is known to
have ideal class number 1. Hence, every 4x4 matrix root of f(x) is similar to
the companion matrix, via a unimodular matrix,
2. Let f (x) = x2 - x + 6. Then ex = f(l ± 1=23) . The field generated by this
has l,a as a basis for its ring of integers. The class number of Z[a] is 3 and
(l,a), (2,1+a), (4,1+a) are representatives of the ideal class group. The corre-
sponding matrix classes have
(~ ~), (-1 2) (-1 4) -4 2' -2 2
as representatives.
311
[The theorem of Latimer and MacDuffee is occasionally used in group theory, e.g. by
Magnus who applied it to the polynomial 2
x +1, or by Plesken, Plesken and Pohst,
who applied it to the cyclic polynomials; Rademacher used it in his work on Dedekind
sums; Reiner used it for integral representation of cyclic groups.]
The link with ideal matrices can be displayed by a theorem and converse.
Theorem 6. Every nXn matrix A satisfying the monic irreducible equation
f (x) = 0 of degree n is of the form where X~ is an ideal matrix for an
ideal ~ in the ideal class corresponding to the matrix class S-lAS.
Proof. The matrix C corresponds to the class of the ideal with Z-basis
n-l l,a, .•. ,a The ideal matrix X~ of an ideal ~ with basis al, ... ,an
ideal class corresponding to A is obtained via
( l) (al) X a. = :
at ~-l ~n a
Hence
hence
CX~l (at) = c( f ) = a( f ) a n-l n-l
n a a
hence
A converse of the preceding theorem is:
312
in the
Theorem 7. Given any Z-matrix root of f(x) = 0 when f (x) is an irreducible
monic polynomial of degree n. Then any Z-matrix X for which x-lAX = C, the
companion matrix of f(x), is an ideal matrix for some ideal ~ in the ideal class
corresponding to A.
Proof. First of all, there is such a Z-matrix X, for A and C are similar
over Q and have the same irreducible (over Q) characteristic polynomial. In fact,
C is the rational canonical form of A. Let be the basis of an ideal
corresponding to A.
We have
(~l) _ (a.l) A. - a . . . a a n n
Let X be a Z-matrix such that X-lAX = C. Hence
hence
+'C'))-+'C')) n n
Since a is a simple characteristic root, this implies that
where k € Q(a). Hence replacing the vector
gives
313
-1 .1 a (a) 1 )
X dn
- C, ' a
hence
Hence
x = XJ{
for an ideal m chosen suitably in its class.
[The last two theorems deal with the similarity of two nxn Z-matrices. In
connection with this note that Suprunenko and D.K. Faddeev have shown independently
that similarity is implied by local similarity over all primes.]
We now consider a generalization of the concept of matrix class if a basis for
the order a different from n-l l,a, •.. ,a is considered -- this includes the case
when an order a does not have any basis of this kind.
Theorem 8. Let be a basis for the order o. Take a representation
of a Ez nXn Z-matrices and let the matrices Al, ..• ,An correspond to the
Then has as characteristic root. There is a 1-1 correspondence
between the classes of such representations under unimodular similarity and the ideal
classes in a and their number coincides with the ideal class number of the order.
Since the ring Z[a] is represented faithfully by Z[A] the matrix class result
appears as a special case.
Proof. Define Z-matrices Xi via
Let Al, ••• ,An be as in the theorem. Then Ai also has wi as a characteristic root:
314
Let 6 be an element of 0 which generates Q(a). Then 6 = I siwi for some
si 6 Z and 6 is a characteristic root of B = L siAi' Again, the corresponding
characteristic vector with components al, ..• ,an can be chosen to be the Z-basis
of an ideal. In this way a correspondence between the ideals classes and classes of
integral matrix representations emerges. Since 6 is a generator, each wi is a
polynomial Pi(6). Hence Pi(6) = Wi is a characteristic root of Pi(B) = Ai with
the same vector Hence
i=l, ... ,n.
In the case of Z[a) we have and where A is a matrix
whose characteristic polynomial has a as a zero and C as companion matrix. Then
A = xcx-l , 6 = a, i-l x
Some special properties of the correspondence discussed in the preceding theorems
will now be mentioned.
Theorem 9. The principal ideal class of Z[a), i.e. Z[a) itself, corresponds
to the matrix class generated by the companion matrix C.
The proof of this theorem is already implicit in the proofs given earlier.
For n ~ 3, in general, all other matrix classes in the order Z [A) contain
special matrices discovered by Ochoa. These matrices have the following appearance
for a's and b's certain rational intergers depending on the polynomial of which a
is a zero:
1 0 o
o 0 o
o 0 1
o 0 o
315
This was proved for the maximal order, but is true for certain other orders too.
Ochoa himself proved these facts under a special assumption. The general state
ment was then obtained by Rehm.
Theorem 10. The ideal class in Z [aJ which corresponds to the class of the
transpose A' Qi A is the complementary ideal class.
For the definition of complementary ideal class, see e.g. Reiner.
Corresponding results can be obtained for the classes of integral nXn Z
representations of z[aJ, under unimodular similarity.
Two results attached to the last theorem are now discussed.
The first one deals with the similarity between A and At.
Since A and A' are similar via an integral matrix S, under all circum
stances, it is of importance to study the nature of such an S. Among other facts
the following holds:
Theorem 11 (Bender, Faddeev, Taussky). Any integral S such that
s-lAS = A'
can be expressed in the form
where )1€Q(a) and form a Z-basis of an ideal in the ideal class
corresponding to A.
The second result concerns the question of symmetric matrix roots in a matrix
class.
If the order obtained from a zero of the polynomial in question is the maximal
order, then the complementary ideal class is the inverse class and then the following
fact follows.
Theorem 12 (Taussky). A symmetric matrix root can lie only in a class which
corresponds to an ideal class of order 1 or 2.
316
Examples: Let F = Q(I4iO). An ideal class of order 2 in this field is
given by (17, 19+1410) with (19 7) 7 19 corresponding to it.
However, not every ideal class of order 2 does correspond to a matrix class
which contains a symmetric matrix, e.g. for the same field the ideal class of
(2, 20 + ..J4iO) .
For rational matrix roots of Z-polynomials, see the work of Faddeev, Sapiro,
Bender.
[Integral representations of orders occur in a basic paper by Zassenhaus whose
title hides this fact. The finiteness of classes is the main issue there. Later
Zassenhaus treated the problem via local considerations. There is further work by
Faddeev concerning quadratic rings. Two recent papers by Plesken deal with integral
representation of orders.l
lIn the connections between algebraic number theory and integral matrices, ideal
theory in number fields gives information on the ring of matrices. A classical
theorem tells that the ring of Z-matrices znxn is a principal ideal ring. (This
was originally obtained by showing that any two nXn matrices have a greatest common
left divisor.) The work of DuPasquier makes it appear as a non commutative euclidean
ring. This was greatly generalized by Chevalley who showed how the ideal structure
in a ring R influences the ideal structure in nXn
R . Another example is a theorem
by L. Levy concerning equivalence of an R-matrix with a block diagonal matrix of
indecomposable blocks. The size of the blocks that can be achieved is bounded by
the class number of the ring (it is well known that for R = Z it is l).l
2. Unimodular matrices and units in algebraic number fields
Several disconnected facts will be mentioned here.
1. Theorem 13 (Taussky). Let the algebraic integer a be a solution of the
monic irreducible equation f (x) = 0 of degree n and A an nXn Z-matrix with
characteristic polynomial f(x). Let E = p(a) be a unit in Z[al where p(x)
is an integral polynomial. Then p(A) is an integral unimodular matrix which
317
commutes with A. Conversely, given a unimodular nXn Z-matrix U which commutes
with A, there are two possibilities:
1) Z[a] is the maximal order. Since U is a polynomial in A it must be an
integral polynomial.
2) Z[a] is not the maximal order. Then p(x) need not be an integral
polynomial.
Proof. The fact that U is a polynomial in A is implied by the irreducibility
of f(x). Then case 1) is obvious in virtue of the isomorphism of Z[a] and Z[A].
Case 2) is demonstrated by the example below.
Example. Let A (1 2) then a = IS. Hence Z[a] is not the maximal order 2 -1
in Q(a). The matrix U ~I + ~A commutes with A, but is not a unit in Z [A].
It corresponds to a unit in the maximal order.
On the other hand, for which has the same characteristic polynomial,
only units in Z[a] emerge, for
(a b) (0 1) = (0 1) (a bd), abc d € Z cdSO SOc '"
implies Sb = c, a= d. This implies that with determinant 2 2 a - Sb = ±l,
qualifies as a unit in Q(A).
Further, the two matrices A considered, although satisfying the same equation,
do not lie in the same matrix class. The order Z[a] has two ideal classes, while
the maximal order in Q(a) has only one.
It is suggested to call integral unimodular matrices which commute with A, but
do not lead to units in Z(a), phantom units.
2. Commuting unimodular matrices
The following theorem was proved by Dade, Malcev, for n= 2 by K. Goldberg.
Theorem 14. A commutative group of unimodular matrices is finitely generated.
The result for n = 2 leads to a statement concerning the modular group: an abelian
subgroup of the modular group is cyclic.
318
The proofs of the theorem are based on the fact that the unit group in an
algebraic number field is finitely generated. In particular in the case n=2, let
A,B be a pair of unimodular matrices with AB=BA. Then it was shown that an
integral e exists such that A = ±em, B = ten. This is connected with the fact
that in a quadratic field the units of infinite order form a cyclic group. [For the
deeper connections emerging in the non-commutative case, see e.g. Bass, Zassenhaus
for modern developments.]
3. Theorem 15 (Dade, D.W. Robinson, Taussky, Ward). A suitable finite power
of a unimodular Q-matrix with integral characteristic roots is integral.
A proof for the 2x2 case will be given. It displays the idea of proof for
the nXn case.
Proof for n=2. Let A be such a matrix. Then its characteristic roots are
units in a quadratic field. Let E be one of these characteristic roots. Then
Z[E] and Z[A] are isomorphic. The order Z[E] in Q(E) has units which form a
cyclic group times fl. A finite power of every unit of the maximal order is a unit
in Z[E]. Further, a multiple of A is integral. Take all the elements of Z[A]
which are integral. They form a suborder of Z[A], hence are isomorphic to a
suborder of Z[E].
A finite power of the units in the order Z[E] is in the suborder isomorphic
with the integral elements in Z[A). By the isomorphism a finite power of A is in
that suborder.
3. The discriminant.
There are two topics concerning the discriminant of an algebraic number field
related to integral matrices which will be discussed.
1. The Dedekind Theorem.
Theorem 16. The discriminant of an algebraic number field is divisible by
all ramified primes and only by them.
2. The discriminant matrix.
1. The proof given here uses the following Lemma (see L.E. Dickson. 108-110;
this lemma was pointed out to the author by D. Maurer).
Lemma. Let wl •...• wn be the basis for an associative separable algebra over
the field F with an identity. An element x = Lxiwi • xi 6 F, is properly nilpotent
(i.e., xy, yx are nilpotent for all y in the algebra) if and only if xl, .•• ,xn
are in the null space of the matrix (trace wiwj ), where trace refers to the
algebra trace.
Proof of Dedekind's Theorem. Let {Wi} be an integral basis for the maximal
order 0 in an algebraic number field. Let p be a rational prime. Consider
O/(p). If (p) splits in 0 into different prime ideals. then O/(p) has no
nilpotent element. hence det (trace w.w.) is not divisible by p. If, however (p) l. J
contains a prime ideal factor ~ to a power larger than 1, then Olp contains an
element which is nilpotent, namely an element divisible by ~,
Hence (trace WiWj ) is singular mod p.
2 but not by 1'"
Work by A. Frohlich on the discriminant is related to this proof.
2. It seems that the matrix for a basis of the
maximal order of an algebraic number field had not been studied previously, only its
determinant. [But in group theory the automorphs of trace forms are studied much
(see Feit). 1
The most immediate question concerning a real quadratic form is its inertia.
The following theorem was obtained.
Theorem 17 (Taussky). Let F be an algebraic number field of degree n with
real and complex conjugate fields. Let be a basis for the
integers. Then the discriminant matrix (trace wiwk) has r l +r2 positive charac-
teristic roots and negative ones.
4. Norms from algebraic number fields and noncommuting matrices.
The following theorem is a sample. It was generalized by Bender to arbitrary
rational matrices.
320
Theorem 18 (Taussky). Let A and S be nXn Q-matrices. Let A have an
irreducible characteristic polynomial f(x) and let
Then
-1 S AS A' (the transpose).
det S
n(n-l) -2- 2
(-1) a norm A
where a€Q, A€Q(a), a a zero of f(x).
For n = 2 this implies det S -norm A.
The n = 2 result can be tied to a commutator of matrices result via the
following theorem.
Theorem 19 (Taussky). Let A, B be 2x2 Q-matrices with A and a as in
Theorem 18. Then
det(AB - BA) -norm A, A€Q(a).
Consider then
det(A- A') = det(A- S-lAS) = det(S-l)det(AS - SA)
-1 = -det(S )normjJ, jJ€Q(a).
Since A-A' is a skew symmetric matrix det(A-A') is a square in Q. Hence the
n=2 case of Theorem 18 follows.
Conversely, it can be shown that Theorem 19 implies the n = 2 case of Theorem
18.
321
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Faddeev, D.K., Introduction to the multiplicative theory of modules of integral
Frobenius automorphism 38, 96, 226 fundamental form 212 fundamental discriminant 138 fundamental units 102
Galois correspondence 188 Galois field 37 Galois group 188 Gauss genera 2, 146 genus characters 150 genus equivalence 2, 145 genus field 176, 228, 233, 242 genus group 146 greatest common divisor (gcd) 8, 26 group extension 236
Hasse conductor theorem 219 Hecke L-function 254 Hensel local theory 208 Hermite basis 62, 76 Hilbert class field 177 Hilbert sequence 195, 216 Hilbert symbol 157 Hilbert Theorem 90 147 Hilbert Theorem 94 179
ideal (definition) 11 ideal classes 49 ideal group 50 imbeddings 109 indecomposables 7 inertial field 196, 199 integral closure 9, 45, 54 integral domain 6
Jacobi symbol 32 Jacobi variety 71
Kronecker basis Kronecker forms Kronecker symbol Kronecker theory
63 210 92 175
*This index does not include the appendices.
327
lattice 20, 114 least common multiple (lcm) 26 Legendre symbol 32 Legendre-Hasse theorem 155 local-global theory 3, 155 local ring 213
Noether axioms 44 norm (absolute) 75, 116 norm (estimates) 128, 131 norm (ideal) 24 norm (number) 74 norm (relative) 74, 192 norm (residue) 157 normed class field theory 231, 259
octahedron 114 orders 79, 83, 143
p-adic numbers 153 parallelepiped 114 p-component 98 Pell's equation 105 primary associate 126 primary discriminants 146 prime factorization 86, 90 prime ideal 43 prime (infinite) 164 prime polynomials 35 prime (ramified) 97 prime (= ax2 + bxy + cy2) 1, 23, 182,
235 primitive form 138 primitive polynomial 35 primitive root 29 principal genus 146 principal ideal 12, 178 principal ideal domain (pid) 13, 50 principalization 177
quadratic field 17, 78, 90, 202, 227 quadratic form 1, 138 quadratic reciprocity 32, 148, 204 quadratic residues 31 quadratic ring (order) 17, 83 quaternionic field 239, 241, 246 quotient field 6
ray class field theory 171 ray class number 169 ray ideal group 169 ray modulus semigroup 164 ray number group 166 regular ideal 89 relative degree 195 relative different 208 relative discriminant 208 relative order 195 relative primeness 8, 14, 26 relative quadratic field 215 representations 261 residue class group 29 Riemann surfaces 67 Riemann zeta-function 133 ring class field 180 ring equivalence 143 ring ideals 89
semi direct product 190 separable 53 simple extension 53 Smith basis 63 strict equivalence 141, 170
Tchebotareff density theorem Tchebotareff monodromy theorem trace 74 transfer of ideals 236, 240
151, 257 206, 241
unique factorization domain (ufd) 7, 35 unit form 211 unit index 252 units 6, 102
volume coordinates 122, 126
Weber-Takagi correspondence 171, 227
zone 19
Fermat's Last Theorem A Genetic Introduction to Algebraic Number Theory
By H. M. Edwards
1977. xv, 410p. cloth (Graduate Texts in Mathematics, V. 50)
This book is essentially an historical presentation of the famous unsolved "Fermat's Last Theorem," and a textbook covering all the major topics of algebraic number theory. It ranges from the work of Fermat through Kummer's theory of "ideal" factorization, in which the theorem is proved for all prime exponents less than 37. More elementary topics, such as Euler's proof of the impossibility of x3 + y3 = Z3, are also presented. Exercises and explicit computations supplement the text.
Number Fields
By D. A. Marcus
1977. viii, 279p. paper (Universitext)
This work presents the classical methods and results of algebraic number theory, in a clear, readable exposition accessible to undergraduates as well as to more advanced students. Relying less heavily on the techniques of localization than most other books in the field, the text explores the subject in depth using algebraic, geometric and analytical methods. Effort is made throughout to keep notation and machinery as simple as possible without sacrificing content.
Number Theory
By H. Hasse
English translation edited by H.G. Zimmer
1978. approx. 680p. 49 iIIus. cloth (Grundlehren der mathematischen Wissenschaften, V. 229)
Hasse's classic work, originally published in 1949, is now available for the first time in English in this newly revised version. The main topic of the book is the foundation of number theory in algebraic number fields and fields of algebraic functions of one variable. In contrast to the ideal-theoretic approach of Oedekind, Hilbert and Noether, Hasse's treatment derives from the work of Kronecker and Kummer, and the valuation theory of his own teacher, Hensel. A recent revival of interest in the methods of Kronecker and Kummer makes this a particularly timely publication.
Basic Number Theory Third Edition
By A. Weil
1974. xviii, 325p. cloth (Grundlehren der mathematischen Wissenschaften, V. 144)
This is a coherent, modern treatment of algebraic number theory by the adele method, followed by a self-contained account (involving no cohomology) of class-field theory. Emphasizing the fundamentals of major numerical theory research and the tools of a generally analytic approach, the book may well be regarded as one of the classics in the field by later historians. This third edition includes a new appendix, "Examples of L-functions."
This three-volume edition contains all of Andre Weirs mathematical work, with the exception of his books. In addition to his published papers, a substantial amount of hitherto unpublished original or inaccessible material has been included. Foremost among the new work is a commentary by Weil himself on his mathematical development and accomplishments. This edition reflects Weirs enormous range of interests and his decisive influence on the course of contemporary mathematics; it is an essential reference source for all mathematicians and math libraries.
Cyclotomic Fields By S. Lang
1978. xi, 253p. cloth (Graduate Texts in Mathematics, V. 59)
Here is the first textbook account of the basic theory of cyclotomic fields. It includes the classical results of Kummer and Stickel berger, and is complemented by the recent insights of Iwasawa, Leopoldt and others. The book should be of great interest to number theorists and algebraic geometers, as well as to topologists who have encountered Bernoulli numbers and polynomials, and to algebraists working in K-theory.
