Leopoldt-type theorems for non-abelian extensions of Q

TL;DR

This paper establishes conditions under which the ring of integers in certain non-abelian Galois extensions of Q, with Galois groups A4, S4, or A5, is free over its associated order, advancing understanding of their Galois module structure.

Contribution

It provides necessary and sufficient conditions for the ring of integers to be free over the associated order in non-abelian Galois extensions with specific groups.

Findings

Criteria for freeness of the ring of integers over the associated order.

Results specific to Galois groups A4, S4, and A5.

Enhanced understanding of Galois module structure in non-abelian extensions.

Abstract

We prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give necessary and sufficient conditions for the ring of integers of K to be free over its associated order in the rational group algebra Q[G].