Admissibility and field relations

Abstract

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.

A note on Galois groups and local degrees

“…In view of Lemma 2.10 of [Nef12], the group H m,p is realisable over any p-adic field. Moreover, from Neukirch's result [Neuk79], we can find a realisation K m /K of G m such that, for every prime v of K m above p, the local extension K m,v /K p has Galois group H m,p .…”