On the restricted Hilbert-Speiser and Leopoldt properties

TL;DR

This paper investigates the conditions under which number fields exhibit Hilbert-Speiser and Leopoldt properties related to Galois extensions, revealing that the converse implications do not always hold and providing specific examples for cyclic groups of prime order.

Contribution

It demonstrates that the converse of the known implication between Leopoldt and Hilbert-Speiser properties fails generally, but holds in many cases, especially for Galois fields with cyclic prime order groups.

Findings

The converse implication fails in general.

The modified version of the converse holds for many Galois fields with cyclic prime order groups.

Examples show the modified converse can hold even when the original does not.

Abstract

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O_L is free as an O_K[G]-module. If O_L is free over the associated order A_{L/K} for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G=C_p has prime order. We give examples with G=C_p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.