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

Abstract

Kawamoto [5, 6] proved that for any prime number $p$ and any $a \in \mathbf{Q}^{\times}$, the cyclic extenstion $\mathbf{Q}(\zeta_p, a^{1/p}) / \mathbf{Q}(\zeta_p)$ has a normal integral basis (NIB) if it is tame. We show that this property is peculier to the rationals $\mathbf{Q}$. Namely, we show that for a number field $K$ with $K \neq \mathbf{Q}$, there exist infinitely many pairs $(p, a)$ of a prime number $p$ and $a \in K^{\times}$ for which $K(\zeta_p, a^{1/p}) / K(\zeta_p)$ is tame but has no NIB. Our result is an analogue of the theorem of Greither et al. [3] on Hilbert-Speiser number fields.