## A note on integral bases of unramified cyclic extensions of prime degree. II.(English)Zbl 0991.11058

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 D. R. Replogle, K. Rubin, A. Srivastav and the reviewer [J. Number Theory 79, 164–173 (1999; Zbl 0941.11044)] and of Carter (forthcoming).

### MSC:

 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11R23 Iwasawa theory

Zbl 0941.11044
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