FIELDS OF DEFINITION FOR ADMISSIBLE GROUPS

Abstract

A finite group $G$ is called admissible over a field $M$ if it is realizable as the Galois group of an extension of $M$ which is contained in a division algebra with center $M$. We consider the extent to which admissibility over $M$ implies admissibility over a subfield $K \subset M$, comparing variations where the division algebra, the extension field, or the Galois extension, are asserted to be dedined over $K$. We completely determine the logical implications between the variants.