ISRAEL JOURNAL OF MATHEMATICS xxx (2013), 1–14

DOI: 10.1007/s11856-013-0018-7

EMBEDDING PROBLEMS WITH LOCAL CONDITIONSAND THE ADMISSIBILITY OF FINITE GROUPS

BY

Nguy˜en Duy Tan

∗

Universitat Duisburg-Essen, FB6, Mathematik, 45117 Essen, Germany

and

Institute of Mathematics, 18 Hoang Quoc Viet, 10307, Hanoi-Vietnam

e-mail: duy-tan.nguyen@uni-due.de

ABSTRACT

Let K be a field of characteristic p > 0, which has infinitely many discrete

valuations. We show that every finite embedding problem for Gal(K) with

finitely many prescribed local conditions, whose kernel is a p-group, is

properly solvable. We then apply this result in studying the admissibility

of finite groups over global fields of positive characteristic. We also give

another proof for a result of Sonn.

1. Introduction

A celebrated theorem of Shafarevich says that every finite solvable group can

be realized as the Galois group of a Galois extension of any given finite alge-

braic number field k. However, Shafarevich’s proof of this theorem is long and

difficult. Seeking a shorter and more conceptual proof of the theorem in the

case of groups of odd order lead Neukirch [Ne1, Ne2] naturally to study em-

bedding problems with local conditions. The papers of Neukirch also play an

important role in studying the admissibility of finite groups, a notion due to

Schacher [Sch]; see, e.g., [N, So2]. See also [HHK, NP] for recent works using

the patching method to study the admissibility of finite groups.

∗ Partially supported by NAFOSTED, the SFB/TR45 and the ERC/Advanced

Grant 226257.

Received August 17, 2012 and in revised form August 31, 2012

1

2 N. D. TAN Isr. J. Math.

Let K be a global field (i.e., a finite extension of the field of rational numbers

Q or of Fq(t), the field of rational functions in one variable over the finite field

with q elements). For every finite embedding problem E for Gal(K) and for each

prime v of K we can associate a local embedding problem Ev for Gal(Kv), where

Kv is the completion of K at v. (Here, for a field L, we denote by Gal(L) the

absolute Galois group of L, i.e., Gal(L) = Gal(Ls/L) where Ls is the separable

closure of L in some algebraically closed closure L of L.) Each global solution

for E then induces naturally local solutions for Ev. Now given a finite collection

of local solutions for Ev, one can ask whether there is a global one which induces

these given local solutions. In the case that k is a number field or that the order

of the kernel of our finite embedding problem is prime to the characteristic of

k, there are! some results which give an affirmative answer to the question. See

[Ne1, Ne2, Ste].

However, to the best of our knowledge, there are not too many results in the

case that the orders of the kernels of the embedding problems are divisible by

the characteristic of K. In this direction, there is a result of Sonn which says

that for any embedding problem 1 → A → E → Gal(M/K) → 1 for a global

field K of characteristic p > 0 where E is a p-group, every given finite set of

local solutions is induced from a global one (see [So1, Theorem 1]).

In this paper, we consider a field k of characteristic p > 0 equipped with

infinitely many discrete valuations and we consider finite embedding problems

for K whose kernels are p-groups. We obtain the following main result, which

says basically that every embedding problem of this form with finitely many

prescribed local conditions has a proper solution. (See Section 3 for definitions.)

Theorem 1.1: Let K be a field of characteristic p > 0, which has infinitely

many discrete valuations. Let S be a finite set of non-equivalent discrete valua-

tions of K and for each v ∈ S we fix an embedding φv : Gal(Kv) → Gal(K) by

fixing a K-embedding K → Kv. Let E : (α : Gal(K) → G, f : Γ → G) be a finite

p-embedding problem for Gal(K). Then for every collection {βv}v∈J of weak so-

lutions to the weak embedding problems (Ev: (α ◦ φv : Gal(Kv)→G, f : Γ → G)),

there is a proper solution β to E and elements nv ∈ N = ker(E) such that

β ◦ φv = inn(nv) ◦ βv for all v ∈ S.

(Here inn(nv) ∈ Aut(Γ) denotes left conjugation by nv.)

This result contains the above-mentioned result of Sonn as a very special case.

Vol. xxx, 2013 EMBEDDING WITH LOCAL CONDITIONS 3

We then use Theorem 1.1 to study the admissibility of finite groups over

global function fields. Let K be a field. Following Schacher [Sch], a finite group

G is called K-admissible if there exists a finite Galois extension L/K with

Galois group isomorphic to G such that L is a maximal commutative subfield of

some finite-dimensional central division algebra over K. We obtain a reduction

theorem for the admissibility of finite groups.

Theorem 1.2: Let K be a global field of characteristic p > 0. Let Γ be a finite

group, P a normal p-subgroup of Γ. If the quotient group Γ/P is K-admissible

then Γ is K-admissible.

We note that Stern, in [Ste, Theorem 1.3], proves the above result under a

stronger assumption that P is a normal Sylow p-subgroup of Γ.

In the last subsection, as another application of Theorem 1.1, we present

another (shorter) proof of a result of Sonn; see Theorem 4.3.

Acknowledgements. The author would like to express his sincere thanks

to Helene Esnault for her support and constant encouragement. The author

thanks Danny Neftin for his many interesting comments and remarks, which

substantially improve the paper, Lior Bary-Soroker for his help, and Moshe

Jarden for his useful comments and suggestions and for sending us his private

note [Ja]. The paper was revised while the author visited the Vietnam Institute

for Advanced Studies in Mathematics (VIASM). The author would like to thank

VIASM for its support and hospitality.

2. Finite-dimensional vector spaces over Fp

The material of this section is taken from [Ja]. We would like to thank Moshe

Jarden for allowing us to do that.

Let K be a field of characteristic p > 0. A polynomial f(X) ∈ K[X ] in one

variable X , with coefficients in K, is called additive if f(x+ y) = f(x) + f(y)

for all x, y in any field extension L of K. It is well-known that an additive

polynomial in one variableX overK is precisely of the form∑m

i=1 biXpi

, bi ∈ K

[La, Chapter VI, §12, page 310].

The additive polynomial f(X) = b0X + b1Xp + · · · + bmX

pm

is separable if

and only if b0 �= 0. In this case

ker f = {x ∈ Ks | f(x) = 0}

4 N. D. TAN Isr. J. Math.

is an additive group annihilated by multiplication with p. The action of Gal(K)

on ker f makes it an Fp[GalK]-module.

Lemma 2.1: Let K be a field of characteristic p > 0.

(a) Let V be a finite Fp-sub-vector-space of K. Then there exists a monic

separable additive polynomial f(X) ∈ K[X ] whose roots are the ele-

ments of V .

(b) Let V be a sub-Fp[Gal(K)]-module of Ks. Then there exists a monic

separable additive polynomial f(X) ∈ K[X ] whose roots are the ele-

ments of V .

Proof of (a): We start from the identity Xp − X =∏p−1

i=0 (X − i) in the ring

K[X ] and substitute XY for X in order to get (XY )p − X

Y =∏p−1

i=0 (XY − i). Then

we multiply the latter identity by Y p to get

(1) Xp −XY p−1 =

p−1∏

i=0

(X − iY ).

In particular, if dim(V ) = 1 and v generates V , then substituting v for Y in (1)

gives

(2) Xp − vp−1X =

p−1∏

i=0

(X − iv).

The left-hand side of (2) is a separable additive polynomial while its right hand

side is a polynomial whose roots are exactly the elements of V .

When dim(V ) = n ≥ 2, we write V = U⊕W with dim(U) = 1 and dim(W ) =

n − 1. An induction hypothesis on n gives a separable additive polynomial

g ∈ K[X ] such that

(3) g(X) =∏

w∈W

(X − w).

Hence, choosing a generator u1 of U , we get from (3) and (2) that∏

v∈V

(X − v) =∏

(u,w)∈U⊕W

(X − u− w) =∏

u∈U

∏

w∈W

(X − u− w)

=∏

u∈U

g(X − u) =∏

u∈U

(g(X)− g(u))

=

p−1∏

i=0

(g(X)− ig(u1)) = g(X)p − g(u1)p−1g(X).

Vol. xxx, 2013 EMBEDDING WITH LOCAL CONDITIONS 5

The choice of u1 implies that u1 �∈ W , so g(u1) �= 0. Hence, the right-hand

side is a separable additive polynomial while the left-hand side is a polynomial

whose roots are exactly the elements of V , as desired.

Proof of (b): The module V is also a finite Fp-sub-vector space of Ks. Hence,

by (a), there exists a monic separable additive polynomial f(X) ∈ Ks[X ] such

that f(X) =∏

v∈V (X − v). By assumption, each σ ∈ Gal(K) permutes the

elements of V . Thus,

(σf)(X) =∏

v∈V

(X − σv) =∏

v∈V

(X − v) = f(X).

It follows that f ∈ K[X ].

Lemma 2.2: Let K be a field of characteristic p and V a finite Fp[Gal(K)]-

module with card(V ) ≤ card(K). Then there exists a monic separable additive

polynomial f(X) ∈ K[X ] such that ker f Gal(K) V .

Proof. Let Λ be the kernel of the action of Gal(K) on V . In other words, Λ

consists of all elements σ ∈ Gal(K) that satisfy σv = v for all v ∈ V . We have

been tacitly assuming that the action of Gal(K) on V is continuous. Since V is

finite, Λ is an open normal subgroup of Gal(K). Therefore, its fixed field L is

a finite Galois extension of K and the action of Gal(K) on V yields a faithful

action of G = Gal(L/K) on V .

We consider both V and K as vector spaces over Fp. By assumption

dimFp(V ) = logp card(V ) ≤ logp card(K) = dimFp K.

Hence, we may inject an Fp-basis of V into K, and extend the injection linearly

to obtain an embedding ψ : V → K as Fp-vector spaces.

Now we use the normal basis theorem to choose a primitive element z for

L/K and define an Fp-linear map ϕ : V → L by the formula

ϕ(v) =∑

σ∈G

ψ(σ−1v)σz.

It turns out that ϕ is an embedding of Fp[G]-modules. Indeed, ϕ is injective

since σz is linearly independent over K and ψ is injective; and for each τ ∈ G

we have

τ(ϕ(v)) =∑

σ∈G

ψ(σ−1v)τσz =∑

ρ∈G

ψ(ρ−1τv)ρz = ϕ(τv).

6 N. D. TAN Isr. J. Math.

By Lemma 2.1, there exists a monic separable additive polynomial f(X) ∈ K[X ]

whose roots are the elements of ϕ(V ). Thus, ker(f) Gal(K) V , as desired.

Remark 2.3: If the field K is infinite, Lemma 2.2 is a special case of a more gen-

eral and deeper result on the structure of linear commutative algebraic groups

which are annihilated by p(= charK); see, e.g., [CGP, Proposition B.1.13] or

[Oe, Chapter V, Proposition 4.1 and Subsection 6.1].

3. p-embedding problems with local conditions

A weak embedding problem E for a profinite group Π is a diagram

E := Π

α

��

Γf

�� G

which consists of profinite groups Γ and G and homomorphisms α : Π → G,

f : Γ → G with f is surjective. (All homomorphsims of profinite groups con-

sidered in this paper are assumed to be continuous.) If in addition α is also

surjective, we call E an embedding problem.

A weak solution of E is a homomorphism β : Π → Γ such that fβ = α. We

say that β is a proper solution if in addition β is surjective. (This forces α

to be surjective, in which case E is an embedding problem.)

We call E a finite weak embedding problem if the group Γ is finite. The

kernel of E is defined to be N := ker(f). We call E a weak p-embedding

problem if N is a p-group.

Let φ1 : Π1 → Π be a homomorphism of profinite groups and β a weak

solution of E . ThenE1 := Π1

α◦φ1

��

Γf

�� G,

is a weak embedding problem and β ◦ φ1 is a weak solution of E1.Suppose that φ = {φj}j∈J is a family of homomorphisms φj : Πj → Π

of profinite groups. We will say that E is weakly (respectively properly) φ-

solvable if for every collection {βj}j∈J of weak solutions to the weak embedding

problems (Ej : (α ◦ φj : Πj → G, f : Γ → G)), there is a weak (respectively

Vol. xxx, 2013 EMBEDDING WITH LOCAL CONDITIONS 7

proper) solution β to E and elements nj ∈ N = ker(E) such that β ◦ φj =

inn(nj) ◦ βj for all j ∈ J .

Let φ = {φj}j∈J be a family of homomorphisms φj : Πj → Π of profinite

groups. We call φ p-dominating (respectively strongly p-dominating) if the

induced map

φ∗ : H1(Π, P ) →∏

j∈J

H1(Πj , P )

is surjective (respectively, surjective with infinite kernel) for every non-trivial

finite elementary abelian p-group P on which Π acts continuously. Recall that

the map φ∗ is given explicitly by the following rule: if g : Π → P is a crossed

homomorphism, then φ∗(g) = (g ◦ φj)j∈J .

We will use the following theorem due to Harbater (see [Ha1, Theorem 2.3]

and [Ha2, Theorem 1]).

Theorem 3.1 (Harbater): Let p be a prime number and let Π be a profinite

group. Consider the following two conditions:

(i) Every finite p-embedding problem for Π has a weak solution (i.e.

cdp(Π) ≤ 1).

(ii) Every finite p-embedding problem for Π is properly φ-solvable, for every

strongly p-dominating family of homomorphisms φ={φj :Πj→Π}j∈J .

Then we have the implication (i) ⇒ (ii).

For basic notions in valuation theory in this note, we refer the reader to [FrJ],

[Ef] and [CF].

Lemma 3.2: Let (K, v) be a henselian valued field with non-trivial value group

Γ. Let f(T ) = b0T + b1Tp + · · · + bmT

pm

be an additive polynomial with

coefficients in K and b0 �= 0. Then there exists γ ∈ Γ such that for every a ∈ K

with v(a) ≥ γ, we have a ∈ f(K).

Proof. Let i0 be an index satisfying v(bi0) = min{v(bi) | i = 0, . . . ,m} and let γ

be an element in Γ such that γ > 2v(b0)−v(bi0) ≥ v(bi0). (The existence of such

an element γ follows from the assumption that Γ is nontrivial.) For any a ∈ K

with v(a) ≥ γ, we prove that a ∈ f(K). In fact, set g(T ) := b−1i0f(T )− b−1

i0a.

Then all coefficients of g(T ) are in OK := {x ∈ K | v(x) ≥ 0} and

v(g(0)) = v(b−1i0a) = v(a)− v(bi0) ≥ γ − v(bi0) > 2v(b0)− 2v(bi0) = 2v(g′(0)).

8 N. D. TAN Isr. J. Math.

Since (K, v) is henselian, g(T ) admits a root x ∈ OK ; see, for example, [Ef,

Theorem 18.1.2 (f)]. It then implies that a = f(x) ∈ f(K).

Lemma 3.3: Let K be a field of characteristic p > 0. Let S be a finite set

of non-equivalent valuations of rank 1 of K. Let f(T ) ∈ K[T ] be a separable

additive polynomial. Then the natural map

K/f(K) →∏

v∈S

Kv/f(Kv)

is surjective, where Kv is the completion of K at v, for each v ∈ S.

Proof. Since Kv is a complete valued field of rank 1, Kv is hensenlian for each

v ∈ S [Ef, Corollary 18.3.2]. So by Lemma 3.2, there exists γv ∈ Γv := v(Kv)

such that for any a ∈ Kv with v(a) ≥ γv, we have a ∈ f(Kv). Therefore,

O(γv) := {x ∈ Kv | v(x) ≥ γv} ⊂ f(Kv).

To show that K/f(K) → ∏vKv/f(Kv), it thus suffices to show that

K →∏

v∈S

Kv/O(γv)

is surjective. But this follows immediately from the Artin–Whaples weak ap-

proximation theorem ([CF, page 48]).

Let K be a field of characteristic p > 0, and v a valuation of K. We will fix

an embedding φv : Gal(Kv) → Gal(K).

Proposition 3.4: Let K be a field of characteristic p > 0. Let S be a fi-

nite set of non-equivalent valuations of K of rank 1. Let φS be the family of

homomorphisms {φv : Gal(Kv) → Gal(K)}v∈S .

(1) The family φS is p-dominating.

(2) If K has a rank-1 valuation w not in S whose value group is non-p-

divisible, then the family φS is strongly p-dominating.

Proof. Let P be a non-trivial elementary p-group on which Gal(k) acts (con-

tinuously). For Part (1), we need to prove that

φ∗S : H1(Gal(K), P ) →

∏

v∈S

H1(Gal(Kv), P )

is surjective, and for Part (2), we need to prove further that φ∗S has an infinite

kernel.

Vol. xxx, 2013 EMBEDDING WITH LOCAL CONDITIONS 9

Consider P as a finite Fp[Gal(K)]-module. Hence by Lemma 2.2, P =

{x ∈ Ks | f(x) = 0}, where f(T ) = b0T + · · · + bmTpm

is an additive poly-

nomial in one variable with coefficients in K with m ≥ 1 and b0bm �= 0. For

each field extension L of K, we have the following exact sequence of Galois

modules

0 −→ P −→ Lsf−→ Ls −→ 0.

From this exact sequence we get the following exact sequence of Galois coho-

mology groups

H0(Gal(L), Ls)f−→ H0(Gal(L), Ls) −→ H1(Gal(L), P ) −→ H1(Gal(L), Ls).

By Hilbert 90, H1(Gal(L), Ls) = 0 (see, e.g., [Se, Chapter II, Proposition 1]),

hence

H1(Gal(L), P ) H0(Gal(L), Ls)/im(f) = L/f(L),

for any field extension L ⊃ K.

In particular, H1(Gal(K), P ) = K/f(K), H1(Gal(Kv), P ) = Kv/f(Kv) and

the map φ∗S becomes the canonical map

ϕS : K/f(K) →∏

v∈S

Kv/f(Kv).

Then the assertion (1) follows immediately from Lemma 3.3.

For the assertion (2), we set S′ = S ∪ {w}. Then, as above, the map

φ∗S′ : H1(Gal(K), P ) →∏

v∈S

H1(Gal(Kv), P )×H1(Gal(Kw), P )

is surjective.

Furthermore, H1(Gal(Kw), P ) is infinite (see [BT, Proof of Theorem 1.1]).

Each element h of H1(Gal(K), P ) such that the first coordinate of φ∗S′ (h) is

zero belongs to the kernel of φ∗S . Thus the kernel of φ∗S is infinite.

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1. Since char(K) = p, we have cdp(Gal(K)) ≤ 1 (see [Se,

Chapter II, Proposition 3]). By Proposition 3.4, part (2), the family φS is

strongly p-dominating. Therefore, by Theorem 3.1, every finite p-embedding

problem for Gal(K) is properly φS-solvable.

Remarks 3.5: (1) For another proof of Lemma 3.2, see [Du, Lemma 2.3].

10 N. D. TAN Isr. J. Math.

(2) Lemma 3.3 could also be proved by noting that f(Kv) is open in Kv

and then using the weak approximation theorem as in the proof of [TT,

Lemma 2].

4. Applications

In this section we will use Theorem 1.1 to prove a reduction theorem on the

admissibility of finite groups over global function fields (Theorem 1.2) and to

give another (shorter) proof of a result of Sonn (Theorem 4.3).

4.1. Admissibility of finite groups over global function fields. To

prove Theorem 1.2, we need the following criterion of admissibility of finite

groups over global fields, which is due to Schacher (see [Sch, Propositions 1, 5

and 6]).

Theorem 4.1 (Schacher): Let K be a global field, G a finite group. Then G

is K-admissible if and only if there exists a finite Galois extension L/K with

Galois group isomorphic toG such that for every prime � dividing the order ofG,

Gal(Lv/Kv) contains an l-Sylow subgroup of G for at least two different primes

v of K, where Kv and Lv are the completions of K and L at v, respectively.

We are now ready to prove Theorem 1.2.

Proof of Theorem 1.2. Set G = Γ/P . By Theorem 4.1, there exists a finite

Galois extension L/K with Galois group Gal(L/K) G such that for every

prime � dividing the order of G, Gal(Lv/Kv) contains an �-Sylow subgroup of

G for at least two different primes v of K.

Let � be any prime number which divides the order of Γ. There are two

discrete valuations v1(�), v2(�) of K such that for each v in {v1(�), v2(�)},Gal(Lv/Kv) contains an �-Sylow subgroup of G. (Note that if � does not divide

the order of G then any �-Sylow subgroup of G is trivial.) Let S be the union of

all {v1(�), v2(�)}, where � runs over the finite set of prime divisors of the order

of Γ.

We consider the following embedding problem

E : Gal(K)

α

��

1 �� P �� Γf

�� G = Gal(L/K) �� 1.

Vol. xxx, 2013 EMBEDDING WITH LOCAL CONDITIONS 11

For each discrete valuation v in S, the local embedding problem:

Ev : Gal(Kv)

αv

��

1 �� P �� Γv

f�� Gv = Gal(Lv/Kv) �� 1,

where Γv = f−1(Gv), has a proper solution, say βv [BT, Theorem 1.1].

By Theorem 1.1, there exists a proper solution β : Gal(K) → Γ of E such that

its induced local solution β|Gal(Kv) : Gal(Kv) → Γ is equal to βv up to an inner

automorphism by an element in P , for each v in S.

Let N/K be the Galois extension corresponding to the solution β. Then

Gal(N/K) = Γ and Gal(Nv/Kv) = Γv.

Let � be any prime divisor of the order of Γ. Let H be any �-Sylow subgroup

of Γ; we prove thatK := f(H) is an �-subgroup of G. It is clear if � �= p, because

in this case H is actually isomorphic to K. If � = p, then P is a subgroup of

H because P is normal in Γ and all p-Sylow subgroups of Γ are conjugate. It

follows that one has the following short exact sequence of finite groups:

1 −→ P −→ Hf−→ K −→ 1.

Thus, |H | = |P |.|K| and hence [G : K] = [Γ : H ] is prime to p. Therefore, K is

a p-Sylow subgroup of G. Note also that f−1(K) = H in all cases.

For v in {v1(�), v2(�)}, Gv contains a conjugate of K. Thus, Γv = f−1(Gv)

contains a conjugate of H , which is an �-Sylow subgroup of Γ. By Theorem 4.1,

Γ is K-admissible.

Inspired by recent works of [HHK, NP] on the admissibility of finite groups,

we would like to raise the following question.

Question 4.2: Does Theorem 1.2 still hold true for other fields of characteristic

p > 0 besides global function fields, e.g.,

(1) a field which is a finitely generated field extension of K of transcendence

degree one, where K is complete with respect to a discrete valuation

and whose residue field is algebraically closed,

(2) the fraction field of a complete local domain of dimension 2, with a

separably closed residue field?

12 N. D. TAN Isr. J. Math.

4.2. A result of Sonn. Let K be a field, G a finite group. In [FS], the

authors say that the pair (K,G) has Property A if every division algebra with

center K and index equal to the order of G is a crossed product for G. They

show that if K is a global field of characteristic not dividing the order of G,

then (K,G) has Property A if and only if G is the direct product of two cyclic

groups Ce and Cf of order e, f , respectively, and K contains the e-th roots of

unity; if K is a global field of finite characteristic p dividing the order of G,

then a necessary condition that (K,G) has Property A is that G has a normal

p-Sylow subgroup with quotient group Ce × Cf , e and f prime to p, and k

contains the e-th roots of unity. In [So1], Sonn shows that the above condition

is also sufficient by proving the following theorem.

Theorem 4.3 ([So1, Theorem 3]): Let K be a global field of characteristic

p > 0, e and f positive integers prime to p, and assume that K contains the

e-th roots of unity. Let Γ be a finite group whose p-Sylow group P is normal,

with factor group G isomorphic to the direct product of two cyclic groups of

order e and f , respectively. Let S be any finite set of primes of K. Then there

exists a Galois extension N of K with Gal(N/K) Γ such that for each v ∈ S,

Gal(Nv/Kv) Γ, where Nv, Kv denote the completion of N , K, respectively,

at v.

Using Theorem 1.1, we can also give another proof of Sonn’s result.

Proof. For each v ∈ S, let Lv be the (tamely ramified) extension ofKv generated

by the unramified extension of kv of degree f and by the e-th root of a prime

element of kv. Since Kv contains the e-th roots of unity, this is an abelian

extension with Galois group isomorphic to G. By Gruenwald–Wang’s theorem

([NSW, Chapter IX, Theorem 9.2.8]), there is a Galois extension L/K with

Gal(L/K) G whose completions coincide with Lv. That means that in the

following diagram,

(E , Ev) : Gal(Kv)� � �� Gal(K)

α

��

1 �� P �� Γf

�� G = Gal(L/K) �� 1,

one has α(Gal(Kv)) = G = Gal(Lv/Kv) (for v in S).

Vol. xxx, 2013 EMBEDDING WITH LOCAL CONDITIONS 13

For each v in S, the embedding problem Ev has a proper solution

βv : Gal(Kv) → Γ (see [BT, Theorem 1.1]). By Theorem 1.1, there exists

a proper solution β : Gal(K) → Γ of E such that its induced local solution

β|Gal(Kv) : Gal(Kv) → Γ is equal to βv up to an inner automorphism by an

element in P , for each v in S.

Let N/K be the Galois extension corresponding to the solution β. Then

Gal(N/K) = Γ and Gal(Nv/Kv) = Γ, as required.

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