A note on integral bases of unramified cyclic extensions of prime degree. II (Q5932140)

This note is concerned with a particular problem in Galois module structure, to wit: Suppose \(L/K\) is a Kummer extension of prime degree \(p\). One knows that existence of a normal integral basis (NIB) implies existence of a power integral basis (PIB), and the question is whether the converse is true. The main result says that the converse fails systematically, that is, there are infinitely many \(K\) admitting a counterexample \(L\), and this for every fixed degree \(N\) of \(K\) over the rationals that is divisible by \(p(p-1)\). The basic method is to exhibit \(K\) containing \(\zeta_p\) and \(\alpha \in E_KK^{*p}/K^{*p}\) such that \(\alpha\) is singular primary but not congruent to 1 modulo \((\zeta_p-1)^p\). If one looks for \(\alpha\) in the ``minus part'' of \(E_KK^{*p}/K^{*p}\), then one can even make do with \(\alpha=\zeta_p\), which suffices for the main result. The authors go on to show (and this is technically more difficult) that in some cases one may also find \(\alpha\) as above in the plus part. This note lends further support to the general idea that \(p\)-Hilbert-Speiser fields \(K\) (that is: fields over which every tame abelian extension of degree \(p\) has NIB) are comparatively rare; see work of \textit{D. R. Replogle, K. Rubin, A. Srivastav} and the reviewer [J. Number Theory 79, 164--173 (1999; Zbl 0941.11044)] and of Carter (forthcoming).