Note on the ring of integers of a Kummer extension of prime degree. V

Abstract

Let $\ell$ be a prime number, and $K$ a number field with $\zeta_{\ell} \in K^{\times}$. We give a simple necessary and sufficient condition for all tame Kummer extensions over $K$ of degree $\ell$ to have a relative normal integral basis. The result is given in terms of the class number and the group of units of $K$.