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.