On solvable number fields

Neukirch, Jürgen

doi:10.1007/bf01390030

Cited by 59 publications

(63 citation statements)

References 14 publications

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“…From a theorem of Shafarevich, this is always possible if k = Q and G is solvable [KM04,theorem 6.1]. Moreover, if G has odd order, one can add the Grunwald conclusion: the completion of F/Q at each prime p ∈ S can be prescribed [Neu79] [NSW08, (9.5.5)]. Here we are interested in ramification prescriptions at finitely many given primes.…”

Section: Introductionmentioning

confidence: 99%

Specialization results and ramification conditions

Legrand

2016

Isr. J. Math.

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Abstract. Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T ) with group G such that E/k is regular, we produce some specializations of E/k(T ) at points t 0 ∈ P 1 (k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior -ramified or unramified -at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T ) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T ).

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Section: Introductionmentioning

confidence: 99%

Specialization results and ramification conditions

Legrand

2016

Isr. J. Math.

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“…For example, a theorem of Neukirch (see [23,Corollary 2]) implies that any group whose order is prime to the number of roots of unity in K is K-preadmissible if and only if it is K-admissible. The Q-preadmissible groups are those with metacyclic Sylow subgroups (see [21]).…”

Section: §4])mentioning

confidence: 99%

Admissibility and field relations

Neftin

2011

Isr. J. Math.

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Abstract. Let K be a number field. A finite group G is called K-admissible if there exists a G-crossed product K-division algebra. K-admissibility has a necessary condition called K-preadmissibility that is known to be sufficient in many cases. It is a 20 year old open problem to determine whether two number fields K and L with different degrees over Q can have the same admissible groups. We construct infinitely many pairs of number fields (K, L) such that K is a proper subfield of L and K and L have the same preadmissible groups. This provides evidence for a negative answer to the problem. In particular, it follows from the construction that K and L have the same odd order admissible groups.

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“…Families of groups G and number fields K for which (G, K, S) has an affirmative answer to the Grunwald problem for every S include: (1) abelian groups of odd order over every number field, by the Grunwald-Wang theorem [5,16]; (2) solvable groups of order prime to the number of roots of unity in K, by Neukirch's theorem [10]; and (3) groups with a generic extension over K, by Saltman's theorem [13].…”

Section: Introductionmentioning

confidence: 99%

The Grunwald problem and approximation properties for homogeneous spaces

Demarche¹,

Arteche²,

Neftin³

2017

Ann. inst. Fourier

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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

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On solvable number fields

Cited by 59 publications

References 14 publications

Specialization results and ramification conditions

Specialization results and ramification conditions

Admissibility and field relations

The Grunwald problem and approximation properties for homogeneous spaces

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