Q-Admissibility of solvable groups

Sonn, Jack

doi:10.1016/0021-8693(83)90085-6

Cited by 23 publications

(23 citation statements)

References 7 publications

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“…Liedahl extended this result to give necessary and sufficient conditions on a pair ÔG, F Õ for G to be F -admissible, where G is a metacyclic p-group and F is a number field ( [Lie96]). In 1983 Sonn settled Problem 11.2 for solvable groups, proving they are all Q-admissible in [Son83].…”

Section: Amitsur Conjecturementioning

confidence: 99%

Open problems on central simple algebras

Auel

Brussel

Garibaldi

et al. 2011

Transformation Groups

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Abstract. We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners. Motivated by these and other developments, we now present an updated list of open problems. Some of these problems are-or have become-special cases of much more general problems for algebraic groups. To keep our task manageable, we (mostly) restrict our attention to central simple algebras and PGL n . The following is an idiosyncratic list that originated during the Brauer group conference held at Kibbutz Ketura in January 2010. We thank the participants in that problem session for their contributions to this text. We especially thank Darrel Haile,

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Section: Amitsur Conjecturementioning

confidence: 99%

Open problems on central simple algebras

Auel

Brussel

Garibaldi

et al. 2011

Transformation Groups

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“…Although still open in general, many particular groups and types of groups satisfying this criterion have been shown in fact to be admissible over Q; see for example [Son83,SS92,CS81,FF90,FV87,Fei93,Fei02,Fei04]. Also, Corollary 10.3 of [Sch68] shows that admissible groups over a global field of characteristic p have metacyclic Sylow subgroups at the primes other than p.…”

Section: Conjecture ([Sch68]) Let G Be a Finite Group Then G Is Admmentioning

confidence: 99%

Patching subfields of division algebras

Harbater

Hartmann

Krashen

2010

Trans. Amer. Math. Soc.

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Abstract. Given a field F , one may ask which finite groups are Galois groups of field extensions E/F such that E is a maximal subfield of a division algebra with center F . This question was originally posed by Schacher, who gave partial results in the case F = Q. Using patching, we give a complete characterization of such groups in the case that F is the function field of a curve over a complete discretely valued field with algebraically closed residue field of characteristic zero, as well as results in related cases.

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“…We note that since the proof of Sonn's theorem ( [24]) over Q is based on Neukirch's [17,Main Theorem] which makes an assumption on the absence of roots of unity in K, Sonn's proof does not apply over arbitrary number fields.…”

Section: Introductionmentioning

confidence: 99%

Tamely ramified subfields of division algebras

Neftin

2013

Journal of Algebra

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For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K = Q, the answer is described by the long standing Q-admissibility conjecture.We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conjecture for solvable groups.

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Q-Admissibility of solvable groups

Cited by 23 publications

References 7 publications

Open problems on central simple algebras

Open problems on central simple algebras

Patching subfields of division algebras

Tamely ramified subfields of division algebras

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