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

TL;DR

This paper investigates how the property of a finite group being K-admissible, related to the existence of certain division algebras, behaves under extensions of number fields, providing criteria for admissibility transfer.

Contribution

It establishes conditions under which K-admissibility extends to larger fields M, especially for Sylow metacyclic and nilpotent groups, using the Liedahl condition.

Findings

K-admissibility often extends under field extensions

Liedahl condition characterizes admissibility over extensions

Results apply to Sylow metacyclic and nilpotent groups

Abstract

A finite group G is K-admissible if there exists 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.