Non-existence and splitting theorems for normal integral bases

TL;DR

This paper investigates conditions under which normal integral bases cannot exist in certain Galois extensions and shows that their existence implies a strong splitting of the extension tower.

Contribution

It introduces new criteria that prevent the existence of normal integral bases and links their existence to the structural splitting of abelian towers.

Findings

New conditions for non-existence of normal integral bases in tame Galois extensions.

Existence of a normal integral basis in an abelian tower implies the tower is strongly split.

Provides technical hypotheses under which these results hold.

Abstract

We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower Q \subset K \subset L forces the tower to be split in a very strong sense.