## Some remarks on Hilbert–Speiser and Leopoldt fields of given type

### Volume 108 / 2007

Colloquium Mathematicum 108 (2007), 217-223
MSC: Primary 11R33.
DOI: 10.4064/cm108-2-5

#### Abstract

Let $p$ be a rational prime, $G$ a group of order $p$, and $K$ a number field containing a primitive $p$th root of unity. We show that every tamely ramified Galois extension of $K$ with Galois group isomorphic to $G$ has a normal integral basis if and only if for every Galois extension $L/K$ with Galois group isomorphic to $G$, the ring of integers $O_L$ in $L$ is free as a module over the associated order ${\cal A}_{L/K}$. We also give examples, some of which show that this result can still hold without the assumption that $K$ contains a primitive $p$th root of unity.