Given a field k of characteristic zero and an indeterminate T over k, we investigate the local behaviour at primes of k of finite Galois extensions of k arising as specializations of finite Galois extensions E/k(T ) (with E/k regular) at points t 0 ∈ P 1 (k). We provide a general result about decomposition groups at primes of k in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study crossed products, the Hilbert-Grunwald property, and finite parametric sets. Recall that the latter is independent of the choice of P (up to k p -isomorphism). 1 2 JOACHIM KÖNIG, FRANÇ OIS LEGRAND, AND DANNY NEFTINExamples of Grunwald problems (G, (L (p) /k p ) p∈S ) with no solution L/k occur already for cyclic groups G, when S contains a prime of k lying over 2 [Wan48]. However, it is expected [Har07, §1] that, for solvable groups G, every Grunwald problem (G, (L (p) /k p ) p∈S ) has a solution, provided S is disjoint from some finite set S exc of "exceptional" primes of k, depending only on G and k. This is known when (1) G is abelian, and S exc is the set of primes of k dividing 2 [NSW08, (9.2.8)]; (2) G is an iterated semidirect product A 1 ⋊ (A 2 ⋊ · · · ⋊ A n ) of finite abelian groups, and S exc is the set of primes of k dividing |G|; see [Har07, Théorème 1] and [DLAN17, Theorem 1.1]; (3) G is solvable of order prime to the number of roots of unity in k, and S exc = ∅ [NSW08, (9.5.5)]; and (4) there exists a generic extension for G over k, and S exc = ∅ [Sal82, Theorem 5.9]. Among the above, the latter is the only method which applies to non-solvable groups. However, the family of non-solvable groups for which a generic extension is known is quite restricted, e.g., it is unknown whether the alternating group A n has a generic extension for n ≥ 6. See [JLY02] for an overview on generic extensions.The main source of realizations of non-solvable groups G over k is via k-regular Gextensions, that is, via G-extensions E/k(T ), where T is an indeterminate over k and k is algebraically closed in E. Indeed, by Hilbert's irreducibility theorem, every non-trivial k-regular G-extension E/k(T ) has infinitely many linearly disjoint specializations E t 0 /k, t 0 ∈ P 1 (k), with Galois group G. Many groups have been realized by this method; see, e.g., [MM99], and references within, as well as [Zyw14] for more recent examples.This specialization process provides a natural way to attack Grunwald problems for k-regular Galois groups, that is, for finite groups G admitting a k-regular G-extension of k(T ). Namely, given such an extension E/k(T ), it is natural to ask for the local behaviour of specializations E t 0 /k, t 0 ∈ P 1 (k). That is, which local extensions L (p) /k p , which local Galois groups Gal(L (p) /k p ), and which local degrees [L (p) : k p ] arise by completing the specialization E t 0 /k at primes p of k, when t 0 runs over P 1 (k)? For points t 0 ∈ P 1 (k) which are p-adically far from branch points of E/k(T ), this approach was deeply investigated by Dèbes ...
Given a group $G$ and a number field $K$, the Grunwald problem asks whether given field extensions of completions of $K$ at finitely many places can be approximated by a single field extension of $K$ with Galois group G. This can be viewed as the case of constant groups $G$ in the more general problem of determining for which $K$-groups $G$ the variety $\mathrm{SL}_n/G$ has weak approximation. We show that away from an explicit set of bad places both problems have an affirmative answer for iterated semidirect products with abelian kernel. Furthermore, we give counterexamples to both assertions at bad places. These turn out to be the first examples of transcendental Brauer-Manin obstructions to weak approximation for homogeneous spaces.Comment: 18 pages. Final version. Accepted for publication in Annales de l'Institut Fourie
For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.On the other hand, It is well known that every finite group G appears as a Galois group of an unramified extension over some number field. This is obtained by realizing G as a Galois group of a tame (tamely ramified) extension L 0 /K over some number field, and finding a (not necessarily Galois) number field M which is disjoint from L 0 and satisfies: "for every prime P of M, the ramification index of P over its restriction p to K is divisible by the ramification index of p in L 0 ". Abhyankar's lemma then implies that L := L 0 M is an unramified extension of M with Gal(L/M) ∼ = G. Moreover, the resulting extension L/M is tamely defined
It is now known that for any prime p and any finite semiabelian p-group G, there exists a (tame) realization of G as a Galois group over the rationals ޑ with exactly d = d(G) ramified primes, where d(G) is the minimal number of generators of G, which solves the minimal ramification problem for finite semiabelian p-groups. We generalize this result to obtain a theorem on finite semiabelian groups and derive the solution to the minimal ramification problem for a certain family of semiabelian groups that includes all finite nilpotent semiabelian groups G. Finally, we give some indication of the depth of the minimal ramification problem for semiabelian groups not covered by our theorem.
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