Realizability and admissibility under extension of $p$-adic and number fields.

Abstract

A finite group $G$ is $K$-admissible if there is a $G$-crossed product $K$-division algebra. In this manuscript we study the behavior of admissibility under extensions of number fields $M/K$. We show that in many cases, including Sylow metacyclic and nilpotent groups whose order is prime to the number of roots of unity in $M$, a $K$-admissible group $G$ is $M$-admissible if and only if $G$ satisfies the easily verifiable Liedahl condition over $M$.