Realizability and Admissibility

under Extension of p-Adic and Number Fields

Danny Neftin and Uzi Vishne^{1}

Received: July 12, 2012

Communicated by Max Karoubi

Abstract. A finite group G is K-admissible if there is a G-crossed product K-division algebra. In this manuscript we study the behavior of admissibility under extensions of number fields M/K. We show that in many cases, including Sylow metacyclic and nilpotent groups whose order is prime to the number of roots of unity in M, a K-admissible group G is M-admissible if and only if G satisfies the easily verifiable Liedahl condition over M.

2010 Mathematics Subject Classification: 16K20, 12F12

Keywords and Phrases: admissible group; adequate field; tame admissibility; Liedahl’s condition.

1. Introduction

Let K be a field. A field L ⊇ K is called K-adequate if it is contained as a maximal subfield in a finite dimensional central K-division algebra. A group G is K-admissible if there is a G-extension L/K, i.e. L/K is a Galois extension with Gal(L/K) ≅ G, so that L is K-adequate. Equivalently, G is K-admissible if there is a G-crossed product K-division algebra. Ever since adequacy and admissibility were introduced in [19], they were studied extensively over various types of fields, especially over number fields. As oppose to realizability of groups as Galois groups, there are known restrictions on the number fields K over which a given group is K-admissible. Liedahl’s condition (which was shown by Schacher [19] over ℚ, and generalized by Liedahl [9, Theorem 28]) describes such a restriction. We say that G satisfies Liedahl’s condition over K, if for every prime p dividing |G|, one of the following holds:

(i) p decomposes in K (has at least two prime divisors), (ii) p does not decompose in K, and a p-Sylow subgroup G(p) of G is metacyclic and admits a Liedahl presentation over K (for details see Definition 2.4 which is based on [9]).

1The second named author is partially supported by a BSF US-Israel grant 2010/149.

In [19, Theorem 9.1], Schacher showed that any finite group G is admissible over some number field K. However, for many groups G it is an open problem to determine the number fields over which they are admissible. In fact, searching for an explicit description for all groups seems hopeless. In this paper we fix a field K over which G is admissible and ask over which finite extensions of K, G is still admissible. We assume our group G is realizable over M and furthermore can be realized over M with prescribed local conditions, i.e. it admits the Grunwald-Neukirch (GN) property. This question then leads (see Section 5.1) via Schacher’s criterion (Theorem 2.1) to the following local realization problem:

Problem 1.1. Let m/k be an extension of p-adic fields and G a group that is realizable over k. Is there a subgroup H of G which is realizable over m and contains a p-Sylow subgroup of G?

At first we consider the case of p-groups, where the problem is whether a pgroup that is realizable over k, is realizable over an extension m of k. For p odd, we notice that the maximal pro-p quotient \overline{G_{k}(p)} of the absolute Galois group G_{k} is covered by \overline{G_{m}(p)}, providing a positive answer:

Proposition 1.2. Let m/k be a finite extension of p-adic fields where p is an odd prime. Then any p-group that is realizable over k is also realizable over m.

The simplest behavior one can hope for in terms of admissibility, is that a Kadmissible group G would be M-admissible if and only if it satisfies Liedahl’s condition over M. This is indeed the case for various classes of groups:

(1) When all Sylow subgroups of G are cyclic [19, Theorem 2.8];

(2) When G is abelian and does not fall into a special case over M [2];

(3) For metacyclic groups [9],[10];

(4) For G = SL_{2}(5) [5];

(5) For G = A_{6} or G = A_{7} [20];

(6) For G = PGL_{2}(7) [1], and

(7) For the Symmetric groups G = S_{n}, 1 ≤ n ≤ 17, n ≠ 12, 13 (by [4], [9]

and [19]). Using Proposition 1.2, we are able to add all the odd-order p-groups having the GN-property over M to this list:

Proposition 1.3. Let M/K be an extension of number fields and p an odd prime. Let G be a p-group that is K-admissible and has the GN-property over M. Then G is M-admissible if and only if G satisfies Liedahl’s condition over M.

Propositions 1.2 and 1.3 are proved in Subsection 3.1. We note (in Section 3) that Proposition 1.3 extends to nilpotent groups of odd order and (by Remark 5.1) to Sylow metacyclic groups (having metacyclic Sylow subgroups). However, the following example shows that Problem 1.1 can have a negative answer for some 2-groups:

Example 1.4. There is a group G, of order 2^{6}, which is realizable over ℚ_{2} but not over ℚ_{2}(\sqrt{−1}).

For a proof, see Corollary 3.6. In Proposition 3.8 we interpret this example globally:

Example 1.5. There is a rational prime q for which the group G of Example 1.4 is ℚ(\sqrt{q})-admissible, satisfies Liedahl’s condition over ℚ(\sqrt{−1}, \sqrt{q}) but is not ℚ(\sqrt{−1}, \sqrt{q})-admissible.

Liedahl showed that a similar phenomena happens for the groups S_{n}, n = 12, 13 and the local extension ℚ_{2}(\sqrt{−3})/ℚ_{2} (see [4]). We shall restrict our discussion to groups which are either of odd order, or with metacyclic 2-Sylow subgroups. For some p-adic extensions m/k, for p odd, including extensions in which the inertia degree f(m/k) is a p-power and [m:k] > 5 we show that G_{m} covers the maximal quotient of G_{k} with a normal p-Sylow subgroup. We use this method to answer Problem 1.1 positively for odd primes, under the following assumptions. The list of ‘sensitive’ extensions of p-adic fields, (16 with p = 3 and one for p = 5) is described in Subsection 4.2.

Theorem 1.6. Let p be an odd prime. Let m/k be a non-sensitive extension of p-adic fields and G a group with a normal p-Sylow subgroup P. Assume G is realizable over k. Then there is a subgroup H ≤ G that contains P and is realizable over m.

The question as to whether the non-sensitivity assumption can be removed remains open. However, the assumption that the Sylow subgroup is normal is essential:

Example 1.7. Let G = C_{7} ≀ D where

D = ⟨a, b | a^{7} = b^{29} = 1, a^{−1}ba = b^{7}⟩;

thus the 7-Sylow subgroups of G are neither normal nor metacyclic. In Example 4.11 we show there exists an extension m/k of 7-adic fields such that G is realizable over k, although no subgroup of G that contains a 7-Sylow subgroup is realizable over m.

We say that an extension of number fields M/K is sensitive if it has a sensitive completion. The main theorem follows from Theorem 1.6 by combining the local data (see Subsection 5.1):

Theorem 1.8. Let M/K be a non-sensitive extension of number fields. Let G be a group for which every Sylow subgroup is either normal or metacyclic, and the 2-Sylow subgroups are metacyclic. Assume G is K-admissible and has the GN-property over M. Then G is Madmissible if and only if G satisfies Liedahl’s condition over M.

Let μ_{n} denote the set of n-th roots of unity. As a consequence of Theorem 1.8 and [13, Corollary 2] we have:

[figure: K(μ_{4q^{u}p^{v}}); K(μ_{q^{u}p^{v}}); K(μ_{4q^{u}}); K(μ_{q^{u}}); K(μ_{4}); K; ℚ]

Figure 1. Admissibility of C_{13} ≀ M_{3^{3}}: the group is admissible over the solid-boxed fields, but not over the dash-boxed ones (see Example 1.11; here u ≥ 2, v ≥ 1, p = 13, q = 3 and μ_{n} are the n-th roots of unity.)

Corollary 1.9. Let G be an odd order group for which every Sylow subgroup is either normal or metacyclic. Let M/K be a non-sensitive extension of number fields so that G is K-admissible and μ_{|G|} ∩ M = {1}. Then G is M-admissible if and only if G satisfies Liedahl’s condition over M. In particular, if every prime dividing |G| decomposes in M or if M ∩ℚ(μ_{|G|}) = K ∩ ℚ(μ_{|G|}), then G is M-admissible.

We also show that the assumption that every Sylow subgroup is either normal or metacyclic is essential in Theorem 1.8:

Example 1.10. Let G be the group defined in Example 1.7. In Example 5.7, we show furthermore that there is an extension of number fields M/K so that G is K-admissible, satisfies Liedahl’s condition over M, has the GN-property over M, but is not M-admissible.

As an example we use Theorem 1.8 to understand the behavior of admissibility for a specific group (see Example 5.4):

Example 1.11. Let K = ℚ(\sqrt{14}) and G = C_{13}≀M_{3^{3}}, where M_{3^{3}} is the modular group

⟨x, y|x^{−1}yx = y^{4}, x^{3} = y^{9} = 1⟩,

and ≀ is the standard wreath product. In Example 5.4, we show G is Kadmissible and deduce from Theorem 1.8 that for a number field M ⊇ K, G is M-admissible if and only if G satisfies Liedahl’s condition. We therefore deduce the admissibility behavior in Figure 1 by checking Liedahl’s condition.

Similar examples (also given in Example 5.4) show that the rank (the minimal number of generators) of the p-Sylow subgroups of K-admissible groups is not bounded (as apposed to the case of admissible p-groups discussed in [19, Section 10]).

The basic facts about admissibility of groups over number fields are reviewed in Section 2. We also discuss the behavior of wild and tame admissibility under extension of number fields and the connection between these types of admissibility to parts (i) and (ii) (respectively) in Liedahl’s condition.

The authors are grateful to Jack Sonn for his valuable advice and useful comments.

2. Preliminaries

2.1. Admissibility and Preadmissibility. For a prime v of a field K, we denote by K_{v} the completion of K with respect to v. If L/K is a finite Galois extension, L_{v} denotes the completion of L with respect to some prime divisor of v in L. The basic criterion for admissibility over global fields is due to Schacher:

Theorem 2.1 (Schacher, [19]). Let L/K be a finite Galois extension of global fields. Then L is K-adequate if for every rational prime p dividing |G|, where G = Gal(L/K), there is a pair of primes v_{1}, v_{2} of K such that each of Gal(L_{v_{i}}/K_{v_{i}}) contains a p-Sylow subgroup of G.

Extracting the necessary local conditions for K-admissibility from Theorem 2.1, we arrive at the following definition. For a group G, G(p) denotes a p-Sylow subgroup.

Definition 2.2. Let K be a number field. The group G is K-preadmissible if G is realizable over K, and there exists a finite set S = {v_{i}(p) : p | |G|, i = 1, 2} of primes of K, and, for each v ∈ S, a subgroup G^{v} ≤ G, such that

(1) v_{1}(p) ≠ v_{2}(p) for every p dividing |G|,

(2) G^{v_{i}(p)} ⊇ G(p) for every p and i = 1, 2, and
(3) G^{v} is realizable over K_{v} for every v ∈ S.

(Notice that a p-group G is K-preadmissible if and only if there is a pair of primes v_{1} and v_{2} of K, such that G is realizable over K_{v_{1}} and over K_{v_{2}}.)

Clearly, every K-admissible group is also K-preadmissible. However the opposite does not always hold (see [11, Example 2.14]). For an extension of fields L/K, Br(L/K) denotes the kernel of the restriction map res : Br(K) → Br(L). For number fields we have the following isomorphism of groups, where Π_{K} is the set of places of K:

Br(L/K) ≅ ( ⨁_{π∈Π_{K}} \frac{1}{gcd_{π′|π} [L_{π′}:K_{π}]}ℤ/ℤ)_{0} ,

where ( ·)_{0} denotes that the sum of invariants is zero. Over a number field K, the exponent of a division algebra is equal to its degree, and so L is K-adequate if and only if there is an element of order [L:K] in Br(L/K) [19, Proposition 2.1].

2.2. Tame and wild admissibility. We denote by k_{un} the maximal unramified extension of a local field k, and by k_{tr} the maximal tamely ramified extension. The tamely ramified subgroup Br(L/K)_{tr} of Br(L/K) is the subgroup of algebras which are split by the tamely ramified part of every completion of L; namely the subgroup corresponding under the above isomorphism to

(2.1)  ( ⨁_{π∈Π_{K}} \frac{1}{gcd_{π′|π} [L_{π′} ∩ (K_{π})_{tr} :K_{π}]}ℤ/ℤ)_{0} .

Following the above local description of adequacy we define:

Definition 2.3. We say that a finite extension L of K is tamely K-adequate if there is an element of order [L:K] in Br(L/K)_{tr}. Likewise, a finite group G is tamely K-admissible if there is a tamely Kadequate Galois G-extension L/K.

The structure of tamely admissible groups is related to Liedahl presentations: For t prime to n, let σ_{t,n} be the automorphism of ℚ(μ_{n})/ℚ defined by σ_{t,n}(ζ) = ζ^{t} for ζ ∈ μ_{n}.

Definition 2.4 ([9]). We say that a metacyclic p-group has a Liedahl presentation over K, if it has a presentation

(2.2)  M(m, n, i, t) := ⟨x, y | x^{m} = y^{i}, y^{n} = 1, x^{−1}yx = y^{t}⟩

such that σ_{t,n} fixes K ∩ ℚ(μ_{n}).

Example 2.5. The dihedral group D_{4} has a Liedahl presentation over ℚ, but not over ℚ(\sqrt{−1}). Thus D_{4} satisfies the Liedahl condition over ℚ, but not over ℚ(\sqrt{−1}).

The existence of Liedahl’s presentation for a p-group G over K implies G is K-tame-preadmissible (namely, Definition 2.2 holds with realizability within (K_{v})_{tr} in 2.2.(3) ).

Remark 2.6 (Liedahl, follows directly from [9, Proofs of theorems 28 and 29]). Let G be a finite group. If G is realizable over infinitely many completions of K (at infinitely many primes), then G has a presentation as above. If G is a p-group then the converse also holds. In addition a p-group is realizable over infinitely many completions of K if and only if it is realizable over a completion K_{v} at one prime v that does not divide p.

This allows us to simplify the definition of preadmissibility by noticing that the primes v_{i}(p), i = 1, 2, p | |G|, in Definition 2.2 can be chosen to be distinct:

Lemma 2.7. Let K be a number field. A group G is K-preadmissible if and only if it is realizable over K, and there are distinct primes v_{i}(p), p runs over the primes dividing |G| and i = 1, 2, such that for every p and i = 1, 2, there is a subgroup H ≤ G that contains a p-Sylow subgroup of G and is realizable over K_{v_{i}(p)}.

Proof. The if part holds by definition. To prove the only if part let

T = {v_{i}(p) | p | |G|, i = 1, 2}

be a set of primes of K and for every prime v ∈ T a corresponding subgroup G^{v} so that

(1) v_{1}(p) ≠ v_{2}(p),

(2) G^{v} is realizable over over K_{v},

(3) G^{v_{i}(p)} contains a p-Sylow subgroup of G, for every p dividing |G| and i = 1, 2. We shall define primes w_{i}(p), p | |G| and i = 1, 2, such that all primes are distinct and for every w_{i}(p) there is a subgroup of G that contains a p-Sylow subgroup of G and is realizable over K_{w_{i}(p)}. If v_{i}(p) divides p define w_{i}(p) = v_{i}(p) for any p | |G| and i = 1, 2. If v_{i}(p) does not divide p then G(p) is metacyclic and has a Liedahl presentation over K (by Remark 2.6). Thus, there are infinitely many primes w of K for which G(p) is realizable over K_{w}. For all primes v_{i}(p) that do not divide p (running over both i and p) choose distinct primes w_{i}(p) which are not in T and for which G(p) is realizable over K_{w_{i}(p)} (such a choice is possible since there are infinitely many such w’s). We have chosen distinct primes w_{i}(p), p | |G| and i = 1, 2, as required. □

Remark 2.8. If a p-group G has a Liedahl presentation over M, then G also has a Liedahl presentation over any subfield K of M.

Theorem 2.9 (Liedahl [9], see also [11]). If G is tamely K-admissible, then G(p) has a Liedahl presentation over K for every prime p dividing |G|.

There are no known counterexamples to the opposite implication. However, the following two results are proved for p-groups in [9, Theorem 30] and in general in [11]:

Theorem 2.10. Let K be a number field and let G be a solvable group with metacyclic Sylow subgroups. Then G is tamely K-admissible if and only if its Sylow subgroups have Liedahl presentations.

Theorem 2.11. Let K be a number field. Let G be a solvable group such that the rational primes dividing |G| do not decompose in K. Then G is Kadmissible if and only if its Sylow subgroups are metacyclic and have Liedahl presentations.

In particular if a solvable group is tamely M-admissible then it is also tamely K-admissible for every K ⊆ M, i.e. tame admissibility goes down for solvable groups. Also, if G is solvable and any prime p | |G| satisfies Item (i) in Liedahl’s condition over M, i.e. does not decompose in M, then G is M-admissible if and only if G is tamely M-admissible. In particular for M = ℚ one has that any solvable group G that is ℚ-admissible is tamely ℚ-admissible. However over larger number fields this is no longer the case. Let us define wild K-admissibility:

Definition 2.12. A G-extension L/K is wildly K-adequate if L/K is Kadequate and there is a prime p dividing |G| such that every prime v of K for which

Gal(L_{v}/K_{v}) ⊇ G(p),

divides p. A K-admissible group G is called wildly K-admissible if every Kadequate G-extension is wildly K-adequate.

Clearly a tamely K-admissible group is not wildly K-admissible. Theorems 2.10 and 2.11 guarantee that a solvable group which is K-admissible but not wildly, is tamely K-admissible. In particular:

Remark 2.13. Every K-admissible p-group which is not tamely K-admissible is wildly K-admissible. So, every non-metacyclic K-admissible p-group is wildly K-admissible.

2.3. The Grunwald-Neukirch (GN) property. A group G has the GNproperty (named after Grunwald and Neukirch) over a number field K if for every finite set S of primes of K and corresponding subgroups G^{v} ≤ G for v ∈ S, there is a Galois G-extension L/K for which Gal(L_{v}/K_{v}) ≅ G^{v} for every v ∈ S. The Grunwald-Wang Theorem shows that except for special cases (see [25]), abelian groups A have the GN-property over K. A large set of examples comes from a Theorem of Neukirch [13, Corollary 2]. Let m(K) denote the number of roots of unity in a number field K.

Theorem 2.14 (Neukirch, [13]). Let K be a number field and G a group for which |G| is prime to m(K). Then G has the GN-property over K.

Another important source of examples is having a generic extension ([18, Theorem 5.9]):

Theorem 2.15 (Saltman). If G has a generic extension over a number field K then G has the GN-property over K.

By [17], if μ_{p} ⊆ K then any group of order p^{3} which is not the cyclic group of order 8 has a generic extension over K. In [16], many groups are proved to have a generic extension over number fields, in particular, any abelian group that does not have an element of order 8. In [16] it is also proved that the class of groups with a generic extension is closed under wreath products. In particular we have:

Corollary 2.16 (Saltman). Let q be an odd prime and let K be a number field that contains the q-th roots of unity. Then any iterated wreath product of odd order cyclic groups and groups of order q^{3} has the GN-property over K.

For more examples see [11]. Under the assumption of the GN-property one has the following characterization of wild admissibility:

Lemma 2.17. Let K be a number field and G a K-admissible group that has the GN-property over K. Then G is wildly K-admissible if and only if there is a prime p_{0} | |G| for which G(p_{0}) does not have a Liedahl presentation over K.

Proof. Assume p_{0} is a prime for which G(p_{0}) does not have a Liedahl presentation over K. Assume on the contrary there is a K-adequate G-extension L/K such that for every p | |G| there is a prime v of K that does not divide p, with Gal(L_{v}/K_{v}) ⊇ G(p). Then G(p_{0}) has a Liedahl presentation over L^{G(p_{0})} and by Remark 2.8 G(p_{0}) has a Liedahl presentation over K, contradiction. On the other hand if all Sylow subgroups have Liedahl presentations then by Remark 2.6 every Sylow subgroup is realizable over infinitely many completions. One can therefore choose distinct primes {v_{i}(p) | i = 1, 2, p | |G|} of K such that G(p) is realizable over K_{v_{i}(p)} and v_{i}(p) |̸ p for every p | |G|, i = 1, 2. Since G has the GN-property it follows that G is tamely K-admissible. □

2.4. Galois groups of local fields. Let k be a p-adic field of degree n over ℚ_{p}. Let q be the size of the residue field \overline{k}, and let p^{s} be the size of the group of p-power roots of unity inside k_{tr}. Then

(1) Gal(k_{un}/k) is (topologically) generated by an automorphism σ, and isomorphic to ℤ̂;

(2) Gal(k_{tr}/k_{un}) is (topologically) generated by an automorphism τ, isomorphic to ℤ̂^{(p′)} (which is the complement of ℤ_{p} in ℤ̂);

(3) The group Gal(k_{tr}/k) is a pro-finite group generated by σ (lifting the above mentioned automorphism) and τ, subject to the single relation σ^{−1}τσ = τ^{q}.

Moreover, σ and τ act on μ_{p^{s}} by exponentiation by some g ∈ ℤ_{p} and h ∈ ℤ_{p}, respectively (Note that g and h are well defined modulo p^{s}). Let \overline{G_{k}(p)} denote the Galois group Gal(m/k), where m is the maximal pextension of k inside a separable closure k̃. Let p^{s_{0}} be the number of roots of unity of p-power order in k. Note that if p^{s_{0}} ≠ 2 then n must be even. The following Theorem summarizes results of Shafarevich [23], Demushkin [3], Serre [21] and Labute [8]:

Theorem 2.18 ([22, Section II.5.6]). When p^{s_{0}} ≠ 2, \overline{G_{k}(p)} have the following presentation of pro-p groups:

\overline{G_{k}(p)} ≅ \{ \begin{matrix}⟨x_{1}, …, x_{n+2} | x_{1}^{p^{s_{0}}} [x_{1}, x_{2}] ⋯ [x_{n+1}, x_{n+2}] = 1⟩, & if s_{0} > 0 \\ ⟨x_{1}, …, x_{n+1}⟩, & if s_{0} = 0 \end{matrix} ;

When p^{s_{0}} = 2 and n is odd,

\overline{G_{k}(p)} ≅ ⟨x_{1}, …, x_{n+2} | x_{1}^{2}x_{2}^{4}[x_{2}, x_{3}] ⋯ [x_{n+1}, x_{n+2}] = 1⟩,

otherwise there is an f ≥ 2 for which \overline{G_{k}(p)} has one of the pro-p presentations:

⟨x_{1}, …, x_{n+2} | x_{1}^{2}[x_{1}, x_{2}]x_{3}^{2^{f}} [x_{3}, x_{4}] ⋯ [x_{n+1}, x_{n+2}] = 1⟩, or
⟨x_{1}, …, x_{n+2} | x_{1}^{2+2^{f}} [x_{1}, x_{2}] ⋯ [x_{n+1}, x_{n+2}] = 1⟩.

Theorem 2.19 (Jannsen, Wingberg, [7], see also [15, Theorem 7.5.10]). The group G_{k} has the following presentation (as a profinite group):

G_{k} = ⟨σ, (τ)_{p′}, (x_{0}, …, x_{n})_{p}^{N} | τ^{σ} = τ^{q}, x_{0}^{σ} = ⟨x_{0}, τ⟩^{g}x_{1}^{p^{s}}[x_{1}, x_{2}] ⋯ [x_{n−1}, x_{n}]⟩ if n is even, and

G_{k} = ⟨σ, (τ)_{p′}, (x_{0}, …, x_{n})_{p}^{N} | τ^{σ} = τ^{q},

x_{0}^{σ} = ⟨x_{0}, τ⟩^{g}x_{1}^{p^{s}}[x_{1}, y_{1}][x_{2}, x_{3}] ⋯ [x_{n−1}, x_{n}]⟩

if n is odd, where (τ)_{p′} denotes that τ is a pro-p′ element (has order prime to p in every finite quotient), and (x_{0}, …, x_{n})_{p}^{N} denotes the condition that the closed normal subgroup generated by x_{0}, ..., x_{n} is required to be a pro-p group. Here, the closed subgroup generated by σ and τ is isomorphic to Gal(k_{tr}/k). The notation ⟨x_{0}, τ⟩ stands for (x_{0}^{h^{p−1}}τx_{0}^{h^{p−2}}τ ⋯ x_{0}^{h}τ)^{\frac{πp}{p−1}} , where π_{p} ∈ ℤ̂ is an element such that π_{p}ℤ̂ = ℤ_{p}. Also, y_{1} is a multiple of x_{1}^{τ^{π_{2}(p+1)}} by an element in the maximal pro-p quotient of the pro-finite group generated by x_{1}, σ^{π_{2}} and τ^{π_{2}}. In particular, in every pro-odd quotient of G_{k}, [x_{1}, y_{1}] is trivial.

Remark 2.20. Notice that G_{k} is a semidirect product of a pro-p group P_{k} and a profinite metacyclic group D_{k}, where P_{k} is the closed normal subgroup generated by x_{0}, …, x_{n} and D_{k} is the closed subgroup generated by σ and τ. The p-Sylow subgroup of G_{k} is therefore the pro-p closure of ⟨σ^{π_{p}}⟩ · P_{k}.

Remark 2.21. If G is admissible over a number field K, then for every p there is a subgroup H ⊇ G(p) which is realizable over a completion of K. In particular, H is a product of a metacyclic group and a normal p-subgroup.

The following result on realizability of metacyclic p-groups will be used in Section 5.

Lemma 2.22. Let k be a p-adic field. Then any metacyclic p-group G is realizable over k.

Proof. Let G = M(m, n, i, t) (see (2.2)). The proof for k ≠ ℚ_{2} is in [12]. For k = ℚ_{2} we cover 2-groups, so m and n are 2-powers and t is odd. In this case \overline{G_{k}(2)} has the pro-2 presentation ⟨a, b, c | a^{2}b^{4}[b, c] = 1⟩ (by Theorem 2.18), i.e. \overline{G_{k}(2)} is isomorphic to the free pro-2 group on three generator modulo the normal closure of the single relation. So the map ϕ : \overline{G_{k}(2)} → G defined by:

a ↦ x^{−2}y^{s}, b ↦ x, c ↦ y,

is well defined (and surjective) whenever (x^{−2}y^{s})^{2}x^{4}[x, y] = 1. As t is odd, t^{2}+1 ≡ 2 (mod 4), \frac{t^{2}+1}{2} is odd and we can choose an s ≡ _{\frac{1}{\frac{t^{2}+1}{2}}} \frac{1}{t^{2}} \frac{1−t}{2}

(mod n) so that s(t^{2} + 1) ≡ \frac{1−t}{t^{2}} (mod n). For such s one has:

(x^{−2}y^{s})^{2}x^{4}[x, y] = x^{−4}y^{st^{−2}+s}x^{4}y^{t−1} = y^{s(t^{−2}+1)t^{4}+t−1} = 1.

Thus ϕ is well defined. □

3. p-groups

A nilpotent group G is K-admissible if and only if all Sylow subgroups of G are K-admissible. In particular studying the behavior of the admissibility of G under extension of number fields is reduced to understanding the behavior of its Sylow subgroups.

3.1. The case p odd. We begin by proving the observation on realizability over extensions of p-adic fields, p odd.

Proof of Proposition 1.2. Let n denote the rank [k:ℚ_{p}] and let t = [m:k]. If t = 1 there is nothing to prove. For t = 2, ([m:k], p) = 1 and hence from a G-extension l/k we can form a G-extension lm/m. Now let t > 2. It suffice to show that \overline{G_{k}(p)} is a quotient of \overline{G_{m}(p)}. By Theorem 2.18, \overline{G_{k}(p)} and \overline{G_{m}(p)} have the following presentations of pro-p groups:

\overline{G_{k}(p)} ≅ \{ \begin{matrix}⟨x_{1}, …, x_{n+2} | x_{1}^{p^{s_{0}}} [x_{1}, x_{2}] ⋯ [x_{n+1}, x_{n+2}] = 1⟩, & if s_{0} > 0 \\ ⟨x_{1}, …, x_{n+1}⟩, & if s_{0} = 0 \end{matrix} ,

and

\overline{G_{m}(p)} ≅ \{ \begin{matrix}⟨x_{1}, …, x_{nt+2} | x_{1}^{p^{s′_{0}}} [x_{1}, x_{2}] ⋯ [x_{nt+1}, x_{nt+2}] = 1⟩, & if s′_{0} > 0 \\ ⟨x_{1}, …, x_{nt+1}⟩, & if s′_{0} = 0 \end{matrix} ,

where p^{s_{0}} and p^{s′_{0}} are the numbers of p-power roots of unity in k and m, respectively. Clearly s_{0} ≤ s′_{0}. Let F_{p}(y_{1}, …, y_{k}) denote the free pro-p group of rank k with generators y_{1}, …, y_{k}. If s′_{0} = 0 then we are done since F_{p}(x_{1}, …, x_{n+1}) is a quotient of F_{p}(x_{1}, …, x_{nt+1}). Suppose s′_{0} > 0. Let ϕ: \overline{G_{m}(p)} → F_{p}(y_{1}, …, y_{\frac{nt+2}{2}} ) be the epimorphism defined by ϕ(x_{2i−1}) = 1 and ϕ(x_{2i}) = y_{i}, i = 1, …, \frac{nt+2}{2}. Now as t > 2 we have:

\frac{nt + 2}{2} = n\frac{t}{2} + 1 ≥ n + 2

and hence there is a projection π:

π: F_{p}(y_{1}, …, y_{\frac{nt+2}{2}} ) → \overline{G_{k}(p)}.

Thus π ∘ ϕ: \overline{G_{m}(p)} → \overline{G_{k}(p)} is an epimorphism. We deduce that every epimorphic image of \overline{G_{k}(p)} is also an epimorphic image of \overline{G_{m}(p)}. □

We can now prove Proposition 1.3. It suffices to prove:

Proposition 3.1. Let M/K be an extension of number fields. Let p be an odd prime and G a p-group that is K-admissible and has the GN-property over M. If G satisfies Liedahl’s condition over M, then G is M-admissible.

Proof. As G is K-admissible, G is realizable over K_{v_{1}}, K_{v_{2}} for two primes v_{1}, v_{2} of K. We claim there are two primes w_{1}, w_{2} of M for which G is realizable over M_{w_{1}}, M_{w_{2}}. Since G has the GN-property over M proving the claim shows G is M-admissible. There are two cases: Case I: p decomposes in M. If one of the primes v_{1}, v_{2} does not divide p, then G is metacyclic and hence by Lemma 2.22, G is realizable over any M_{w_{1}}, M_{w_{2}} for any two primes w_{1}, w_{2} of M that divide p. If on the other hand both v_{1}, v_{2} divide p then by Proposition 1.2, G is realizable over M_{w_{i}} for w_{i}|v_{i}, i = 1, 2, Case II: p does not decompose in M. Since G satisfies Liedahl’s condition over M, G has a Liedahl presentation over M. In particular by Theorem 2.11, G is M-admissible. □

As a corollary we deduce that wild admissibility goes up for p-groups:

Corollary 3.2. Let p be an odd prime. Let M/K be an extension of number fields and G a wildly K-admissible p-group that has the GN-property over M. Then G is also wildly M-admissible.

Proof. Since G is wildly K-admissible, p decomposes in K and hence in M. Thus G satisfies Liedahl’s condition over M and by Proposition 3.1 G is Madmissible. By remarks 2.6 and 2.13, the wild K-admissibility of G implies that G is not realizable over K_{v} for any prime v which does not divide p. By Remark 2.8, G is also not realizable over M_{w} for any prime w of M which does not divide p. Therefore an M-adequate G-extension must also be wildly M-adequate and hence G is wildly M-admissible. □

Apply Theorem 2.14, we have:

Corollary 3.3. Let p be an odd prime. Let M/K be an extension of number fields so that M does not contain the p-th roots of unity. Let G be a p-group that is wildly K-admissible. Then G is wildly M-admissible.

3.2. The case p = 2. As in Proposition 1.2 we have:

Lemma 3.4. Let m/k is a finite extension of 2-adic fields, which is either of degree greater than 2, or such that m and k contain \sqrt{−1} and have the same 2-power roots of unity. Then any 2-group realizable over k is also realizable over m.

Proof. If [m:k] > 2, the same proof as of Proposition 1.2 holds in all cases of Theorem 2.18. If \sqrt{−1} ∈ k, and k and m have the same number of 2-power roots of unity then \overline{G_{k}(p)} and \overline{G_{m}(p)} have the same type of presentations in Theorem 2.18 and one can obtain an epimorphism: \overline{G_{m}(p)} → \overline{G_{k}(p)} simply by dividing by the redundant generators of \overline{G_{m}(p)}. □

However, we show that Proposition 1.2 may fail for p = 2. We begin with some group-theoretic preparations.

Lemma 3.5. The group

G = ⟨a_{1}, a_{2}, a_{3} | G′ is central of exponent 2, a_{1}^{2} = [a_{2}, a_{3}], a_{2}^{2} = a_{3}^{2} = 1⟩

is not a quotient of the pro-2 group

Γ = ⟨x_{1}, …, x_{4} | x_{1}^{4}[x_{1}, x_{2}][x_{3}, x_{4}] = 1⟩.

Proof. For j, k = 1, 2, 3, write α_{jk} = [a_{j}, a_{k}] ∈ G. Suppose x_{i} ↦ a_{1}^{t_{i,1}}a_{2}^{t_{i,2}}a_{3}^{t_{i,3}}z_{i} (i = 1, …, 4) is an epimorphism Γ → G, where z_{i} ∈ G′. Then [x_{2i−1}, x_{2i}] ↦ ∏_{j,k=1}^{3}[a_{j}^{t_{2i−1,j}}, a_{k}^{t_{2i,k}}] = ∏_{1≤k<j≤3} α_{jk}^{t_{2i−1,j}t_{2i,k}−t_{2i−1,k}t_{2i,j}}. Since exp(G) = 4, the defining relation of Γ translates to

∏_{i=1}^{2} ∏_{1≤k<j≤3} α_{jk}^{t_{2i−1,j}t_{2i,k}−t_{2i−1,k}t_{2i,j}} = 1,

from which it follows that t_{1,j}t_{2,k} − t_{1,k}t_{2,j} + t_{3,j}t_{4,k} − t_{3,k}t_{4,j} ≡ 0 (mod 2) for every 1 ≤ k < j ≤ 3. Let V denote the vector space 𝔽_{2}^{4}, endowed with the bilinear form b: V ×V → 𝔽_{2} defined by b((v_{i})_{i=1}^{4}, (v′_{i})_{i=1}^{4}) = v_{1}v′_{2} −v_{2}v′_{1} +v_{3}v′_{4} −v_{4}v′_{3}. This is an alternating non-degenerate form (in fact, hyperbolic), and letting t^{j} ∈ V be the vectors t_{i}^{j} = t_{i,j}, we have that b(t^{j}, t^{k}) = 0 for every j, k = 1, 2, 3. It follows that T = span \{t^{1}, t^{2}, t^{3}\} ⊂ V is orthogonal to itself. But then dim T ≤ \frac{1}{2} dim V = 2, contradicting the assumption that the induced map Γ → G/G′ = C_{2}^{3} is surjective. □

Corollary 3.6. There is a group of order 2^{6} which is realizable over k = ℚ_{2} but not over m = ℚ_{2}(\sqrt{−1}).

Proof. As before, we construct a quotient of \overline{G_{k}(2)} which is not a quotient of \overline{G_{m}(2)}. Let G and Γ be as in Lemma 3.5. By Theorem 2.18, \overline{G_{m}(2)} ≅ Γ and

\overline{G_{k}(2)} = ⟨x_{1}, x_{2}, x_{3} | x_{1}^{2}x_{2}^{4}[x_{2}, x_{3}] = 1⟩.

Mapping x_{i} ↦ a_{i} projects \overline{G_{k}(2)} onto G, which is not a quotient of Γ. □

It seems that 2^{6} is the minimal possible order for such a 2-group.

Remark 3.7. Let m/k be an extension of local fields. If there is one 2-group which is realizable over k but not over m, then there are infinitely many such groups. Indeed, let G be such a group, and let k′ be a G-Galois extension of k; the Galois group of any 2-extension of k′ which is Galois over k has G as a quotient, and so is not realizable over m.

Let us apply this example to construct an extension of number fields M/K for which the group G of Lemma 3.5 is wildly K-admissible but not M-admissible and not even M-preadmissible. Let p and q be two primes for which:

1) p ≡ 5 (mod 8)

2) q ≡ 1 (mod 8)

3) q is not a square mod p.

Proposition 3.8. Let K = ℚ(\sqrt{q}), M = K(i) and G be the group from Lemma 3.5. Then G is wildly K-admissible but not M-preadmissible.

Proof. Since G is a 2-group that is not metacyclic it is realizable only over completions at primes dividing 2. In particular if G is K-admissible then G is wildly K-admissible. As 2 splits in K, any (of the two) prime divisor v of 2 in M has a completion M_{v} ≅ ℚ_{2}(i). By Corollary 3.6, G is not realizable over ℚ_{2}(i) and hence not M-preadmissible. It therefore remains to show G is K-admissible. The rational prime p is inert in K. Let 𝔭 be the unique prime of K that divides p. We have N(𝔭) := |\overline{K_{𝔭}}| = p^{2} ≡ 1 (mod 8). Thus, K_{𝔭} has a totally ramified C_{8}-extension. Let 𝔮_{1}, 𝔮_{2} be the two primes of K dividing 2. Consider the field extension L_{0} = K(μ_{8}, \sqrt{p})/K. It has a Galois group

Gal(L_{0}/K) ≅ C_{2}^{3} ≅ G/Z(G).

This extension is ramified only at 𝔮_{1}, 𝔮_{2} and 𝔭. As K_{𝔮_{i}} ≅ ℚ_{2} and Gal((L_{0})_{𝔮_{i}}/K_{𝔮_{i}}) ≅ C_{2}^{3}, (L_{0})_{𝔮_{i}}/K_{𝔮_{i}} is the maximal abelian extension of K_{𝔮_{i}} of exponent 2, for i = 1, 2. Note that N(𝔭) ≡ 1 (mod 8) and hence K_{𝔭} contains μ_{8}. Notice that Z(G) = G′ ≅ C_{2}^{3} and G/Z(G) ≅ C_{2}^{3}. Let us show that the central embedding problem

(3.1)  \xymatrix{ &  &  & G_{K} \ar[d] \ar@{-->}[dl] \\ 0 \ar[r] & Z(G) \ar[r] & G \ar[r] & G/Z(G) \ar[r] & 0,}

has a solution. Let π denote the epimorphism G → G/Z(G). By theorems 2.2 and 4.7 in [14], there is a global solution to Problem 3.1 if and only if there is a local solution at every prime of K. There is always a solution at primes of K which are unramified in L_{0} so it suffices to find solutions at 𝔭, 𝔮_{1}, 𝔮_{2}. Any G-extension of K_{𝔮_{i}} contains (L_{0})_{𝔮_{i}} (as it is the unique C_{2}^{3} extensions of ℚ_{2}), i = 1, 2. Since G is realizable over ℚ_{2} we deduce that the induced local embedding problem (3.2)

\xymatrix{ & G_{K_{𝔮_{i}}} \ar[d] \ar@{-->}[dl] \\ π^{−1}(Gal((L_{0})_{𝔮_{i}}/K_{𝔮_{i}})) = G \ar[r] & Gal((L_{0})_{𝔮_{i}}/K_{𝔮_{i}}) ≅ G/Z(G) \ar[r] & 0,}

has a solution for i = 1, 2. Since (L_{0})_{𝔭} is the ramified C_{2}-extension of K_{𝔭}, it can be embedded into the totally ramified C_{4}-extension and hence the local embedding problem at 𝔭 has a solution. Therefore, Embedding problem 3.1 has a solution. Let L be the corresponding solution field. As Problem 3.1 is a Frattini embedding problem such a solution must be surjective globally and at {𝔮_{1}, 𝔮_{2}}. Thus, L_{0}/K can be embedded in a Galois G-extension L/K for which Gal(L_{𝔮_{i}}/K_{𝔮_{i}}) ≅ G, for i = 1, 2. The field L is clearly K-adequate and hence G is K-admissible. □

4. Realizability under extension of local fields

Realizability of a group G as a Galois group over a field k is clearly a necessary condition for k-admissibility. When k is a local field, the conditions are equivalent since a division algebra of index n is split by every extension of degree n. In this section we study realizability of groups under field extensions, assuming the fields are local.

4.1. Totally ramified extensions. We first note what happens under prime to p local extensions:

Lemma 4.1. Let G_{1} be a subgroup of G that contains a p-Sylow subgroup of G and is realizable over the p-adic field k. Let m/k be a finite extension for which ([m:k], p) = 1. Then there is a subgroup G_{2} ≤ G_{1} that contains a p-Sylow subgroup of G and is realizable over m.

Proof. Let l/k be a G_{1}-extension. Then lm/m is a Galois extension with Galois group G_{2} which is a subgroup of G_{1} and for which [l : l ∩ m] = |G_{2}|. Since ([l ∩ m:k], p) = 1, any p^{s} | [l:k] = |G_{1}| also divides p^{s} | [l:l ∩ m] = |G_{2}|. Thus G_{2} must also contain a p-Sylow subgroup of G. □

The case where p divides the degree [m:k] is more difficult. Let us consider next totally ramified extensions:

Lemma 4.2. Let p ≠ 2. Let G be a group, k a p-adic field with n = [k : ℚ_{p}] and m/k a totally ramified finite extension. Assume furthermore that m/k is not the extension ℚ_{3}(ζ_{9} + \overline{ζ}_{9})/ℚ_{3}. If G is realizable over k then G is also realizable over m.

Remark 4.3. This shows that if G has a subgroup G_{1} that contains a p-Sylow of G and is realizable over k then we can pick G_{2} := G_{1} as a subgroup which contains a p-Sylow subgroup of G and is realizable over m.

Proof. Let m/k be a totally ramified extension of degree r = [m:k]. We shall construct an epimorphism G_{m} → G_{k}. For this we shall consider the presentations given in Theorem 2.19. Denote the parameters of k by n, q, s, g and h. Then the degree of m over ℚ_{p} is nr and its residue degree remains q. Denote the rest of the parameters over m by s′, g′ and h′ (the parameters that correspond to s, g and h in Theorem 2.19). Then by Theorem 2.19, G_{m} has the following presentation (as a profinite group):

G_{m} = ⟨σ, (τ)_{p′}, (x_{0}, …, x_{nr})_{p}^{N} |

τ^{σ} = τ^{q}, x_{0}^{σ} = ⟨x_{0}, τ⟩^{g′}x_{1}^{p^{s′}} [x_{1}, x_{2}] ⋯ [x_{nr−1}, x_{nr}]⟩,

if nr is even and

G_{m} = ⟨σ, (τ)_{p′}, (x_{0}, …, x_{nr})_{p}^{N} |

τ^{σ} = τ^{q}, x_{0}^{σ} = ⟨x_{0}, τ⟩^{g′}x_{1}^{p^{s′}} [x_{1}, y_{1}][x_{2}, x_{3}] ⋯ [x_{nr−1}, x_{nr}]⟩, if nr is odd. Let P_{k} be the closed normal subgroup of G_{k} generated by x_{0}, …, x_{n} and let D_{k} (resp. D_{m}) be the closed subgroup generated by σ and τ. By assumption, P_{k} is a pro-p group. Note that as k and m have the same residue degree (same q), D_{k} ≅ D_{m}. Let us construct the epimorphism from G_{m} to G_{k}. First send x_{0} and every x_{k} with k odd in the presentation of G_{m} to 1. We get an epimorphism

G_{m} ↠ ⟨σ, (τ)_{p′}, (z_{1}, …, z_{d})_{p}^{N} | στσ^{−1} = τ^{q}⟩

where d = ⌈\frac{nr−1}{2}⌉ and ⌈γ⌉ denotes the smallest integer ≥ γ. Let us continue under the assumption d ≥ n + 1. Then there is an epimorphism F_{p}(d) ↠ F_{p}(n + 1). We therefore obtain epimorphisms:

G_{m} ↠ ⟨σ, (τ)_{p′}, (z_{1}, …, z_{n+1})_{p}^{N} | στσ^{−1} = τ^{q}⟩ ↠ D_{k} ⋉ P_{k} ↠ G_{k}.

The numerical condition ⌈\frac{nr−1}{2}⌉ ≥ n + 1 fails if and only if:

(1) r = 1, or

(2) r = 2, or
(3) r = 3 and n = 1.

The case r = 1 is trivial. Since p is an odd prime, the cases r = 2 and r = 3 are done by Lemma 4.1, unless p = 3 and [m:k] = r = 3, in which case n = 1, so k = ℚ_{3}. If m ≠ ℚ_{3}(ζ_{9} + \overline{ζ}_{9}) then m ∩ k_{tr} = k and the parameters g, h, s in the presentation of G_{k} remain the same in the presentation of G_{m}. In such case there is an epimorphism from G_{m} onto G_{k} whose kernel is generated by ⟨x_{2}, x_{3}⟩. □

So the case k = ℚ_{3} and m = ℚ_{3}(ζ_{9} + \overline{ζ}_{9}) remains open. This will be one of several sensitive cases.

4.2. The sensitive cases.

Definition 4.4. We call the extension m/k sensitive if it is one of the following: (1) k = ℚ_{3} and m is the totally ramified 3-extension ℚ_{3}(ζ_{9} + \overline{ζ}_{9}),

(2) k = ℚ_{5} and m = ℚ_{5}(ζ_{11}) is the unramified 5-extension,

(3) [k:ℚ_{3}] = 1, 2, 3 and m/k is the unramified 3-extension,

(4) k = ℚ_{3} and m = ℚ_{3}(ζ_{7}) is the unramified 6-extension.

Remark 4.5. There are 17 sensitive field extensions, up to isomorphism: one over ℚ_{5} and 16 over ℚ_{3}. Fixing the algebraic closures of the respective p-adic fields, there is one sensitive 5-adic extension and 27 3-adic ones. This can be verified using the automated tools in [6]. We provide details in the Appendix.

Let us formulate the problem in case (1) for odd order groups:

Remark 4.6. Given a field F denote by G_{F}^{odd} the Galois group that corresponds to the maximal pro-odd Galois extension of F. In the p-adic case, for odd p, this is obtained from the presentation of G_{k} (see Theorem 2.19) simply by dividing by the 2-part of σ and τ. In such case we get a presentation of G_{F}^{odd} by identifying σ_{2} = τ_{2} = 1. We get that y_{1} is a power of x_{1} and hence [x_{1}, y_{1}] = 1.

Question 4.7. Let m/k be the sensitive extension (1). Then q = 3; also p^{s} = 3 so we can choose h = −1. For m we have p^{s_{m}} = 9 and τ(ζ_{9} + ζ_{9}^{−1}) = ζ_{9} + ζ_{9}^{−1}, so h_{m} = −1 as well. Theorem 2.19 gives us the presentations:

G_{k}^{odd} = ⟨σ, (τ)_{3′}, (x_{0}, x_{1})_{3}^{N} | τ^{σ} = τ^{3}, x_{0}^{σ} = ⟨x_{0}, τ⟩x_{1}^{3}⟩,

while:

G_{m}^{odd} = ⟨σ, (τ)_{3′}, (x_{0}, x_{1}, x_{2}, x_{3})_{3}^{N} | τ^{σ} = τ^{3}, x_{0}^{σ} = ⟨x_{0}, τ⟩x_{1}^{9}[x_{2}, x_{3}]⟩,

where σ, τ are of order prime to 2 and ⟨x_{0}, τ⟩ = (x_{0}τx_{0}^{−1}τ)^{\frac{π}{2}} , which has order a power of 3 in every finite quotient. Does the following hold: Let G be an epimorphic image of G_{k}^{odd}, is there necessarily a subgroup G(p) ≤ G_{0} ≤ G so that G_{0} is an epimorphic image of G_{m}^{odd}?

Note that for a 3-group G the claim was proved in Proposition 1.2.

Remark 4.8. In fact quotients of G_{k}^{odd} with τ = 1 can be covered: the group ⟨σ, (x_{0}, x_{1})_{3}^{N} | x_{0}^{σ} = x_{1}^{3}⟩ = ⟨σ, (x_{1})_{3}^{N}⟩ is covered by σ ↦ σ, τ ↦ 1, x_{0} ↦ 1, x_{1} ↦ 1, x_{2} ↦ x_{1} and x_{3} ↦ 1. This corresponds to realization of G over k whose ramification index is a 3-power.

4.3. Extensions of local fields. We can now approach the general case. Recall the presentation of G_{k} from Theorem 2.19. Let P_{k} denote the closed normal subgroup generated by x_{0}, …, x_{n} and D_{k} the closed subgroup generated by σ and τ, as in Remark 2.20.

Remark 4.9. (1) Decompose ⟨σ⟩ into its p-primary part generated by σ_{p} and its complement generated by σ_{p′} so that σ = σ_{p}σ_{p′}, where [σ_{p}, σ_{p′}] = 1. Then the pro-p closure of ⟨σ_{p}⟩ · P_{k} is a p-Sylow subgroup of G_{k}.

(2) In every finite quotient σ_{p} is a power of σ, and so normalizes τ. It follows that [τ, σ_{p}] is a power of τ, and so a pro-p′ element.

(3) The image of the closure of ⟨σ_{p}⟩P_{k} is normal in a quotient of G_{k} if and only if τ conjugates σ_{p} into the closure of ⟨σ_{p}⟩P_{k}; but then the image of [τ, σ_{p}] is a pro-p element, so by (2) this is the case if and only if the image of [τ, σ_{p}] is trivial.

(4) Let Ḡ_{k} denote the maximal quotient of G_{k} with a normal p-Sylow subgroup. It follows from (3) that Ḡ_{k} is defined by the relation [τ, σ_{p}] = 1.

Lemma 4.10. Let p be an odd prime. Let m/k be an extension of p-adic fields with f = [m̄:k̄] a p-power, and ⌈\frac{nr−1}{2}⌉ ≥ n+2 where n = [k:ℚ_{p}] and r = [m:k]. Then Ḡ_{k} is a quotient of G_{m}.

Proof. We construct an epimorphism from G_{m} to Ḡ_{k}. Let s_{m}, g_{m}, h_{m} be the invariants s, g, h in Theorem 2.19 that correspond to m, and let n = [k:ℚ_{p}]. Theorem 2.19 gives the following presentation of G_{m}:

G_{m} = ⟨σ, (τ)_{p′}, (x_{0}, …, x_{nr})_{p}^{N} |

τ^{σ} = τ^{q^{f}} , x_{0}^{σ} = ⟨x_{0}, τ⟩^{g_{m}}x_{1}^{p^{sm}} [x_{1}, x_{2}] ⋯ [x_{nr−1}, x_{nr}]⟩, if nr is even and

G_{m} = ⟨σ, (τ)_{p′}, (x_{0}, …, x_{nr})_{p}^{N} | τ^{σ} = τ^{q^{f}} ,
x_{0}^{σ} = ⟨x_{0}, τ⟩^{g_{m}}x_{1}^{p^{sm}} [x_{1}, y_{1}][x_{2}, x_{3}] ⋯ [x_{nr−1}, x_{nr}]⟩,

if nr is odd. Let P_{k} (resp. P_{m}) be the closed normal subgroup generated by x_{0}, …, x_{n} (resp. x_{0}, …, x_{nr}) in G_{k} (resp. G_{m}) and let D_{k} ≤ G_{k} (resp. D_{m} ≤ G_{m}) be the closed subgroup generated by σ, τ in G_{k} (resp. in G_{m}). Set d = ⌈\frac{nr−1}{2}⌉, so by assumption d ≥ n + 2. Similarly to Lemma 4.2 (noting that this time D_{m} can be viewed as a subgroup of index f in D_{k}), we have an epimorphism

G_{m} ↠ ⟨σ, (τ)_{p′}, (z_{1}, …, z_{m})_{p}^{N} | τ^{σ} = τ^{q^{f}} ⟩
≅ ⟨(σ_{p})_{p}, (σ_{p′})_{p′}, (τ)_{p′}, (z_{1}, …, z_{m})_{p}^{N} | τ^{σ_{p}σ_{p′}} = τ^{q^{f}} , [σ_{p}, σ_{p′}] = 1⟩.

Let us divide by the relations z_{m}^{f} = σ_{p} and τ^{z_{m}} = τ^{qσ_{p′}^{−1/f}}, where σ_{p′}^{−1/f} is well defined since f is a p-power. We then obtain an epimorphism to

⟨(σ_{p′})_{p′}, (τ)_{p′}, (z_{1}, …, z_{m})_{p}^{N} | τ^{z_{m}} = τ^{qσ_{p}^{′−1/f}} , [z_{m}^{f}, σ_{p′}] = 1⟩.

Adding the relation [z_{m}, σ_{p′}] and sending z_{m} ↦ σ_{p} maps this group onto

⟨(σ_{p′})_{p′}, (τ)_{p′}, (z_{1}, …, z_{m−1}, σ_{p})_{p}^{N} | τ^{σ_{p}σ_{p′}^{1/f}} = τ^{q}, [σ_{p}, σ_{p′}] = 1⟩.

Mapping σ_{p′} ↦ σ_{p′}^{f}, this groups maps onto

⟨(σ_{p})_{p}, (σ_{p′})_{p′}, (τ)_{p′}, (x_{0}, ⋯, x_{n})_{p}^{N} | τ^{σ_{p}σ_{p′}} = τ^{q}, [σ_{p}, σ_{p′}] = [σ_{p}, τ] = 1⟩,

since by Remark 4.9 the assumption that the normal subgroup generated by σ_{p} is a p-group is equivalent to [σ_{p}, τ] = 1. But Ḡ_{k} is a quotient of this group by Theorem 2.19 and Remark 4.9.(4). □

Using Lemma 4.10, we can prove:

Proof of Theorem 1.6. Let n, q be as defined above for k, and let r = [m:k]. Let f = [\overline{m}:\overline{k}] = f_{p}f_{p′} where f_{p} is a p-power and f_{p′} is prime to p. There is an unramified C_{f}-extension m′/k which lies in m, and then m/m′ is totally ramified. Denote by m_{p} the subfield of m′ which is fixed by C_{f_{p}}. Let r′ = \frac{r}{f_{p′}} = [m:m_{p}]. By Lemma 4.1, there is a subgroup G_{0} ≤ G that contains a p-Sylow subgroup of G and an epimorphism ϕ : G_{m_{p}} → G_{0}. The list of sensitive cases satisfies that if m/k is non-sensitive and m_{p}/k is unramified and prime to p, then m/m_{p} is also non-sensitive and therefore we can assume without loss of generality that m_{p} = k, G = G_{0}, i.e f_{p′} = 1, f = f_{p}, and r′ = r. If ⌈\frac{nr−1}{2}⌉ ≥ n + 2 then G is a quotient of G_{m} by Lemma 4.10. This numerical condition fails if and only if

(1) r = 4, 5 and n = 1;

(2) r = 3 and n = 1, 2, 3;
(3) r = 1, 2.

The cases r = 1, 2, 4 are covered by Lemma 4.1. We are left with cases r = 3, 5. For r = 5, n = 1 so k = ℚ_{5} and by Lemma 4.2 we may assume m/k is not totally ramified, so m/k is the unramified 5-extension which is sensitive. Let r = 3. Lemma 4.1 covers the case p ≠ 3, so we may assume p = 3. Note that f | r. If f = 1, Lemma 4.2 applies, except for m = ℚ_{3}(ζ_{9} + \overline{ζ_{9}}) and k = ℚ_{3}, which is sensitive. If f = 3, then m/k is the unramified 3-extension and n = [k : ℚ_{3}] = 1, 2, 3 which are all sensitive. □

The following example shows that the assumption in Theorem 1.6 that the normal p-Sylow subgroup of G is normal, is essential.

Example 4.11. Let p < q be odd primes such that p^{p} ≡ 1 (mod q) and p ≢ 1 (mod q) (for example p = 7 and q = 29). Let G = C_{p} ≀ D where

D = ⟨a, b | a^{p} = b^{q} = 1, a^{−1}ba = b^{p}⟩.

Let k = ℚ_{p} and m/k the unramified extension of degree p. Then G is realizable over k but there is no subgroup of G that contains a p-Sylow subgroup of G and is realizable over m.

Proof. Let P = C_{p}^{pq}, so that G = P ⋊ D. Then one has the projections

G_{k} → G_{k}^{odd} = ⟨σ, (τ)_{p′}, (x_{0}, x_{1})_{p}^{N} | τ^{σ} = τ^{p}, x_{0}^{σ} = ⟨x_{0}, τ⟩x_{1}^{p}⟩ →
→ ⟨σ, (τ)_{p′}, (x_{1})_{p}^{N} | τ^{σ} = τ^{p}, x_{1}^{p} = 1⟩,

where the latter two group are pro-odd and the second epimorphism is obtained by dividing by x_{0}. The latter group maps onto G by σ ↦ a and τ ↦ b. It is therefore left to prove that for any homomorphism ϕ : G_{m} → G, Im(ϕ) does not contain a p-Sylow subgroup of G. Assume on the contrary that H = Im(ϕ) does. Recall:

G_{m} = ⟨σ, (τ)_{p′}, (x_{0}, …, x_{p})_{p}^{N} |

τ^{σ} = τ^{p^{p}}, x_{0}^{σ} = ⟨x_{0}, τ⟩x_{1}^{p}[x_{1}, y_{1}][x_{2}, x_{3}] ⋯ [x_{p−1}, x_{p}]⟩.

Since q is the only prime dividing |G| other than p, and τ is pro-p′, any map into G must split through:

G_{m} → ⟨σ, (τ)_{p′}, (x_{0}, …, x_{p})_{p}^{N} | [σ, τ] = 1, τ^{q} = 1,
x_{0}^{σ} = ⟨x_{0}, τ⟩x_{1}^{p}[x_{1}, y_{1}][x_{2}, x_{3}] ⋯ [x_{p−1}, x_{p}]⟩.

However the latter group has a normal p-Sylow subgroup which is the product of the closed normal subgroup generated by the x_{i}’s and the pro-p group generated by σ^{π_{p}}. In particular, letting π: G → G/P = D be the projection, the image of πϕ has a normal p-Sylow subgroup. This implies πϕ is not surjective. But H contains a p-Sylow subgroup of G, so we must have Im(πϕ) = C_{p}. Again since H contains a p-Sylow subgroup, and in particular P, we must have H = P ⋊ C where C = C_{p} is a subgroup of D and the action of C on P is induced from the action of D. Thus:

rank(H) = rank(H/[H, H]) = rank((P/[P, C]) × C) = q + 1.

Since H is a p-group any epimorphism to it must split through \overline{G_{m}(p)}. However rank(\overline{G_{m}(p)}) = [m:k] + 1 = p + 1, leading to a contradiction. □

5. Extensions of number fields

We shall now apply Theorem 1.6 to study admissibility and wild admissibility.

5.1. Main Theorem.

Proof of Theorem 1.8. As mentioned in the introduction, Liedahl’s condition is necessary. Let us show that if G satisfies this condition then G is M-admissible. We claim that one can choose distinct primes w_{i}(p), i = 1, 2, p | |G|, of M and corresponding subgroups H_{i}(p) ≤ G so that H_{i}(p) contains a p-Sylow subgroup of G and is realizable over M_{w_{i}(p)}, i = 1, 2. As G is K-admissible, for every p | |G| there are two options:

(1) there are two primes v_{1}, v_{2} of K dividing p and two subgroups G(p) ≤ G_{i} ≤ G so that G_{i} is realizable over K_{v_{i}}, i = 1, 2.

(2) G(p) is realizable over K_{v} for v which does not divide p. In case (1) with p odd and G(p) normal in G, by Theorem 1.6, for any prime w dividing v_{1} or v_{2} there is a subgroup G(p) ≤ H_{w} ≤ G that is realizable over M_{w}. Choose two such primes w_{1}(p), w_{2}(p) and set H_{i}(p) := H_{w_{i}(p)} (the subgroups Theorem 1.6 constructs). In case G(p) is not normal or p = 2, we assumed G(p) is metacyclic and by Lemma 2.22, G(p) is realizable over any M_{w} for any prime w dividing v_{1} or v_{2}. In such case similarly choose two such primes w_{1}(p), w_{2}(p) and set H_{i}(p) = G(p). In case (2), G(p) is metacyclic. If p has more than one prime divisor in M then there are two primes w_{1}(p), w_{2}(p) so that w_{i}(p) divides p and by Lemma 2.22 H_{i}(p) := G(p) is realizable over M_{w_{i}(p)}, i = 1, 2. If p has a unique prime divisor in M then G(p) is assumed to have a Liedahl presentation. Liedahl’s condition implies that there are infinitely many primes w(p) for which G(p) is realizable over M_{w(p)} (see Theorem 28 and Theorem 30 in [9]). Thus, we can choose two primes w_{1}(p), w_{2}(p) for every prime p | |G| that has only one prime divisor in M, so that the primes w_{i}(p), i = 1, 2, are not divisors of any prime q | |G| and are all distinct. For such p, we also choose H_{i}(p) := G(p). We have covered all cases of behavior of divisors of rational primes in M and hence proved the claim. It follows that G is M-preadmissible and as G has the GN-property over M, G is M-admissible. □

Remark 5.1. If G has metacyclic Sylow subgroups, the proof of Theorem 1.8 does not use Theorem 1.6 and holds for sensitive extensions as well.

5.2. Wild admissibility. As to wild admissibility Theorem 1.8 and Lemma 2.17 give:

Corollary 5.2. Let M/K be a non-sensitive extension. Let G be a Kadmissible group for which every Sylow subgroup is either normal or metacyclic and the 2-Sylow subgroups are metacyclic. Assume G has the GN-property over M, satisfies Liedahl’s condition over M but there is a prime p for which G(p) does not have a Liedahl presentation over M. Then G is wildly M-admissible.

We deduce that for groups as in Theorem 1.8, wild admissibility goes up in the following sense that generalizes Corollary 3.2:

Corollary 5.3. Let M/K be a non-sensitive extension. Let G be a wildly Kadmissible group for which every Sylow subgroup is either normal or metacyclic and the 2-Sylow subgroups are metacyclic. Assume G has the GN-property over K and M satisfies Liedahl’s condition over M. Then G is wildly M-admissible.

Proof. By Theorem 1.8, G is M-admissible. The assertion now follows from Lemma 2.17, applied to both K and M, and the fact that if G(p) does not have a Liedahl presentation over K then G(p) does not have a Liedahl presentation over M (see Remark 2.8). □

5.3. Examples. The following is an example in which Theorem 1.8 is used to understand how admissibility behaves under extensions of a given number field:

Example 5.4. Let p, q be odd primes and m an integer so that m is not square mod q but is a square mod p and p ≡ q + 1 (mod q^{2}). For example p = 13, q = 3, m = 14. Let K = ℚ(\sqrt{m}) and G = C_{p} ≀ H, where H is one of the following groups:

(1) H = M_{q^{3}} is the modular group of order q^{3}, i.e.

H = ⟨x, y|x^{−1}yx = y^{q+1}, x^{q} = y^{q^{2}} = 1⟩.
(2) H = C_{pq} × C_{q}.
(3) H = C_{t} where t ∈ ℕ_{odd} is prime to p.

We shall show in each of the cases G is K-admissible. Let M be any nonsensitive extension of K. By Theorem 2.14, in case (1) G satisfies the GN-property over any number field that does not have any p-th and q-th roots of unity, in particular over K. By Corollary 2.16, in cases (2),(3), G satisfies the GN-property over any M and in case (1) if M contains the q-th roots of unity. In cases (2),(3), G satisfies Liedahl’s condition over any M and in case (1), G satisfies Liedahl’s condition over M if and only if q decomposes in M or M ∩ ℚ(μ_{q^{2}}) ⊆ ℚ(μ_{q}). It follows from Theorem 1.8, that in cases (2) and (3) G is M-admissible. In case (1), if one assumes M does not contain any p-th and q-th roots of unity or that M contains the q-th roots of unity then G is M-admissible if and only if G satisfies Liedahl’s condition.

Proof. The prime p splits (completely) in K. Denote it’s prime divisors in K by v_{1}, v_{2}. Then K_{v_{i}} ≅ ℚ_{p} for i = 1, 2. Using the presentation of G_{ℚ_{p}}^{odd} given in Question 4.7 and dividing by x_{0} = 1 one obtains an epimorphism:

(5.1)  G_{ℚ_{p}} → ⟨σ, (τ)_{p′}, (x_{1})_{p}^{N} | τ^{σ} = τ^{p}, x_{1}^{3} = 1⟩.

Since p ≡ q + 1 (mod q^{2}) there is an epimorphism

⟨σ, (τ)_{p′} | τ^{σ} = τ^{p}⟩ → M_{q^{3}}

which together with Epimorphism 5.1 shows that C_{p} ≀ M_{q^{3}} is an epimorphic image of G_{ℚ_{p}}. The group G in case 3 can obtained as an epimorphic image of G_{ℚ_{p}} after dividing 5.1 by τ = 1. In case 2, since q|p−1, there is an epimorphism

⟨σ, (τ)_{p′} | τ^{σ} = τ^{p}⟩ → C_{pq} × C_{q},

which together with 5.1 can be used to construct an epimorphism onto G. In particular C_{p} ≀ H is realizable over K_{v_{1}}, K_{v_{2}} in all cases. Since M_{q^{3}}, C_{q} × C_{q} and C_{t} have Liedahl presentations over K, they are realizable over completions at infinitely many primes of K. As G has the GN-property over K, it follows that G is K-admissible in all cases. □

Remark 5.5. As Case 3 of Example 5.4 shows, the rank of p-Sylow subgroups of K-admissible groups is not bounded as apposed to the case of admissible p-group in which the rank of the group is bounded (see [19, Section 10]).

Remark 5.6. Case 2 in Example 5.4 is an example of a group for which proving M-admissibility requires the use of all steps in the proof of Theorem 1.6.

The following example shows that the assumption that every Sylow subgroup is either normal or metacyclic is essential for Theorem 1.8 even for odd order groups and non-sensitive extensions:

Example 5.7. As in Example 4.11, let p < q be odd primes such that p^{p} ≡ 1 (mod q) and p ≢ 1 (mod q). Let G = C_{p} ≀ D where

D = ⟨a, b | a^{p} = b^{q} = 1, a^{−1}ba = b^{p}⟩.

Let d be a non-square integer that is a square mod pq and K = ℚ(\sqrt{d}). Let v_{1}, v_{2} be the primes of K dividing p. Let M/K be a C_{p}-extension in which both v_{1} and v_{2} are inert and M does not contain any p-th and q-th roots of unity. Since both p and q have more than one prime divisor in M, G satisfies Liedahl’s condition over M. As M does not have any p-th and q-th roots of unity, by Theorem 2.14, G has the GN-property over M and K. We shall now show G is K-admissible but not M-admissible. Note that the only condition of Theorem 1.8 that fails is that either G has a normal p-Sylow subgroup or a metacyclic one.

Proof. By Example 4.11, G is realizable over ℚ_{p} and hence over K_{v_{1}}, K_{v_{2}}. As G has the GN-property over K, G is K-admissible. On the other hand, since G(p) is not metacyclic, a subgroup of G that contains G(p) is realizable only over completions of M at prime divisors of p. Let w_{i} be the prime dividing v_{i} in M, i = 1, 2. Then w_{1}, w_{2} are the only primes dividing p in M but by Example 4.11 G is not realizable over M_{w_{i}}, for i = 1, 2. In particular G is not M-preadmissible and not M-admissible. □

Appendix

We use the Jannsen-Wingberg presentation of G_{ℚ_{3}} to count the sensitive extensions, as defined in Subsection 4.2, up to isomorphism. There is a single extension in each of cases (1), (2) and (4). In case (3) the extension is unramified, so it suffices to count the ground field k, which we do by degrees over ℚ_{3}. In degree 1 there is one case. In degree 2 there are |ℚ_{3}^{×}/ℚ_{3}^{×}^{2}|−1 = 3 quadratic extensions. Since the abelianization of \overline{G_{ℚ_{3}}(3)} is C_{3}^{2}, there are \frac{3^{2}−1}{2} = 4 Galois cubic extensions of ℚ_{3}. For every non-Galois cubic extension k there is a unique S_{3}- Galois extension of ℚ_{3} (generated over k by the square root of the discriminant). The S_{3}-Galois extensions of ℚ_{3} are in one-to-one correspondence to the normal subgroups of G_{ℚ_{3}} with quotient S_{3}. The number of such subgroups is the number of epimorphisms from G_{ℚ_{3}} to S_{3}, divided by |Aut(S_{3})| = 6. When counting epimorphisms φ: G_{ℚ_{3}} → S_{3}, we may assume the generators are in S_{3}, which simplifies the presentation a great deal. Since p^{s} = q = 3 and we may assume g = 1 and h = −1, the presentation is

G_{ℚ_{3}} = ⟨σ, (τ)_{3′}, (x_{0}, x_{1})_{3}^{N} | τ^{σ} = τ^{3}, x_{0}^{σ} = (x_{0}τx_{0}^{−1}τ)^{\frac{π}{2}} x_{1}^{3}[x_{1}, y_{1}]⟩.

However, since x_{1} is a pro-3 element, we may assume x_{1}^{3} = 1. Since all elements of order 3 in S_{3} commute with each other, we may assume [x_{1}, y_{1}] = 1 (see Theorem 2.19 for more details on y_{1}). Since τ is a pro-3′ element, φ(τ) has order at most 2, so φ(x_{0}τx_{0}^{−1}τ) is a commutator, whose order must divide 3. Exponentiation by \frac{π}{2} squares such elements. So every epimorphism to S_{3} splits through

⟨σ, (τ)_{3′}, (x_{0}, x_{1})_{3}^{N} | τ^{2} = x_{0}^{3} = x_{1}^{3} = 1, τ^{σ} = τ^{3}, x_{0}^{σ} = (x_{0}τx_{0}^{−1}τ)^{2}⟩,

and we count epimorphisms from this group. If φ(τ) = 1 then φ(x_{0}) = 1, so φ(x_{1}) is a non-trivial element of order 3 and φ(σ) ∉ ⟨φ(x_{1})⟩. There are 6 such epimorphisms. So assume φ(τ) is a non-trivial involution. The relations give [φ(σ), φ(τ)] = 1, so φ(σ) is either 1 or φ(τ). Moreover, it turns out that φ(x_{0}τx_{0}^{−1}τ)^{2} = φ(x_{0}) whenever φ(x_{0}) has order dividing 3. For each possible value of φ(τ) we get 8 epimorphisms with φ(σ) = 1 and 2 more with φ(σ) = φ(τ). There are 3 involutions, providing us with 6 + 3 · 10 = 36 epimorphisms all together. Dividing by the number of automorphisms, we have 6 Galois extensions of ℚ_{3} with Galois group S_{3}. Each Galois extension of this type contains 3 non-Galois cubic extensions of ℚ_{3}. The S_{3}-extension is determined by the cubic extension, being its Galois closure. So we have 3 · 6 = 18 non-Galois cubic subfields of a fixed algebraic closure ℚ̄_{3}, consisting of 6 isomorphism classes. Summing up, there are 1 + 1 + (1 + 3 + (4 + 6)) + 1 = 17 sensitive field extensions, up to isomorphism.

References

[1] E. Allman, M. Schacher, Division algebras with PSL(2, q)-Galois maximal subfields. J. Algebra 240 (2001), no. 2, 808–821.

382 Danny Neftin and Uzi Vishne

[2] J. Charbit, On p-groups and K-admissibility. Master Thesis supervised by J. Sonn, Technion, Haifa, (2005).

[3] S. P. Demuškin, The group of a maximal p-extension of a local field. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 329-346.

[4] Z. Girnius, S. Liedahl, K-admissibility of S_{12}, S_{13}, S_{14}, S_{15}. J. Algebra 185 (1996), 357–373.

[5] P. Feit, W. Feit, The K-admissibility of SL(2, 5). Geom. Dedicata, 36 (1990), no. 1, 1–13.

[6] W. J. Jones, D. P. Roberts, Database of local fields,

   http://math.la.asu.edu/∼jj/localfields/.

[7] U. Jannsen, K. Wingberg, Die Struktur der absoluten Galoisgruppe p-adischer Zahlkörper. Invent. Math. 70 (1982/83), no. 1, 71-98.

[8] J. P. Labute, Classification of Demushkin groups. Canad. J. Math. 19 (1967), 106-132.

[9] S. Liedahl, Presentations of metacylic p-groups with applications to K-admissibility questions. J. Algebra 169 (1994), no. 3, 965-983.

[10] S. Liedahl, K-admissibility of wreath products of cyclic p-groups. J.Number Theory 60, (1996) 211–232.

[11] D. Neftin, Admissibility and Realizability over Number fields. Submitted. See also:

   arXiv:0904.3772v1.

[12] D. Neftin, Admissibliity as an equivalence relation. Submitted arXiv:0910.4156.

[13] J. Neukirch, On solvable number fields. Invent. Math. 53 (1979), no. 2, 135–164.

[14] J. Neukirch, Über das Einbettungsproblem der algebraischen Zahlentheorie. Invent. Math. 21 (1973), 59-116.

[15] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 323. Springer-Verlag, Berlin (2000).

[16] D. J. Saltman, Generic Galois extensions. Proc. Natl. Acad. Sci. U.S.A. 77(1980), 3, part 1, 1250-1251.

[17] D. J. Saltman, Galois groups of order p^{3}. Comm. Algebra 15 (1987), no. 7, 1365–1373.

[18] D. J. Saltman, Generic Galois extensions and problems in field theory. Adv. in Math. 43 (1982), no. 3, 250–283.

[19] M. Schacher, Subfields of division rings. I. J. Algebra 9 (1968) 451-477.

[20] M. Schacher, J. Sonn, K-admissibility of A_{6} and A_{7}. J. Algebra 145 (1992), no. 2, 333–338.

[21] J.-P. Serre, Structure de certains pro-p-groupes (d’apres Demuskin). Sem. Bourbaki 252 (1962-63)

[22] J.-P. Serre, Galois Cohomology, Springer, edition of 1996.

[23] I. R. Shafarevič On p-extensions. (Russian. English summary) Rec. Math. Mat. Sb 20(62) (1947), 351-363.

[24] L. Stern, On the admissibility of finite groups over global fields of finite characteristic. J. Algebra 100 (1986), no. 2, 344-362.

[25] S. Wang, On Grunwald’s theorem. Ann. of Math. (2) 51 (1950), 471-484.

Danny Neftin 530 Church Street, Ann Arbor, MI 48109, USA neftind@tx.technion.ac.il Uzi Vishne Department of Mathematics Bar Ilan University Ramat Gan 52900 Israel vishne@math.biu.ac.il