On the restricted Hilbert–Speiser and Leopoldt properties

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 $\mathcal{O}_L$ is free as an $\mathcal{O}_K[G]$-module. If $\mathcal{O}_L$ is free over the associated order $\mathcal{A}_{L/K}$ 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 $K/\mathbb{Q}$ Galois) when $G=C_{p}$ has prime order. We give examples with $G=C_5$ to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.