Normal integral bases for cyclic Kummer extensions

Let $K/F$ be a Kummer cyclic extension of number fields. In the case when the degree is a prime number, G'omez Ayala gave an explicit criterion for the existence of a normal integral basis. More recently Ichimura proposed a generalization of that result for cyclic extensions of arbitrary degree, but we have found that Ichimura's result is incorrect. In this paper we present a counter-example to Ichimura's result as well as the correct generalization of G'omez Ayala's result