Admissibility and Realizability over Number Fields

Details Admissibility and Realizability over Number Fields Admissibility and Realizability over Number Fields

2009-04-24

arxiv.org/abs/0904.3772v1

Mathematics

Rings and Algebras

33 pages

Scientific paper

Let K be a number field. A finite group G is K-admissible if there is a K-division algebra with a (maximal) subfield L for which Gal(L/K)= G. The method that was used in most proofs of K-admissibility was to satisfy the local conditions in Schacher's criterion and then find a global realization satisfying these local conditions. We shall see that this approach works in the cases of tame admissibility (in particular when L is tamely ramified over K) of solvable groups, admissibility of most of the abelian groups and admissibility of some larger classes of groups. Many conjectures regarding K-admissibility are based on the guess that the K-admissible groups are those that satisfy the local conditions. We shall construct an example of a special case in which there is an abelian 2-group A and a number field K for which A satisfies the local conditions but A is not K-admissible.

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