Hilbert-Speiser number fields and the complex conjugation

Abstract

Let $p$ be a prime number. We say that a number field $F$ satisfies the condition $(H_p)$ when any tame cyclic extension $N/F$ of degree $p$ has a normal integral basis. We determine all the CM Galois extensions $F/\mathit{Q}$ satisfying $(H_p)$ for the case $p\ge 5$, using the action of the complex conjugation on several objects associated to $F$.