Admissible groups over number fields

Deependra Singh
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Abstract

Given a field K, one may ask which finite groups are Galois groups of field extensions L/K such that L is a maximal subfield of a division algebra with center K. This connection between inverse Galois theory and division algebras was first explored by Schacher in the 1960s. In this manuscript we consider this problem when K is a number field. For the case when L/K is assumed to be tamely ramified, we give a complete classification of number fields for which every solvable Sylow-metacyclic group is admissible, extending J. Sonn's result over the field of rational numbers. For the case when L/K is allowed to be wildly ramified, we give a characterization of admissible groups over several classes of number fields, and partial results in other cases.
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数域上的可容许群
给定一个域 K,我们可能会问,哪些有限群是域扩展 L/K 的伽罗瓦群,从而使 L 成为以 K 为中心的除法代数的最大子域?在本手稿中,我们考虑的是 K 为数域时的问题。对于假定 L/K 完全夯化的情况,我们给出了一个完整的数域分类,对于这些数域,每个可解的 Sylow-metacyclic 群都是可容许的,从而扩展了 J. Sonn 在有理数域上的结果。对于允许 L/K 任意横切的情况,我们给出了几类数域上可容许群的特征,并给出了其他情况下的部分结果。
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