Admissibility and field relations

TL;DR

This paper constructs pairs of number fields with different degrees that share the same preadmissible groups, providing evidence against the possibility that such fields can have different admissible groups.

Contribution

It constructs infinitely many pairs of number fields where one is a subfield of the other and they share the same preadmissible groups, addressing a 20-year open problem.

Findings

Constructed infinitely many pairs of fields with same preadmissible groups

Showed that such fields can have different degrees over Q

Provided evidence against the uniqueness of admissible groups for different fields

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.