On a normal integral bases problem over cyclotomic \(\mathbb{Z}_p\)-extensions
This paper studies the Kummer extensions \(K(\alpha^{(1/p)})\) of a number field which are unramified and possess a relative normal basis (RNB). The main topic is the study of the situation where \(K\) varies amongst the levels of a cyclotomic \(\mathbb{Z}_p\) extension. Using the complex conjugation, one can break up the problem into two parts: \textit{J. Brinkhuis} [Manuscr. Math. 75, 333-347 (1992; Zbl 0757.11044)] has already proved that the minus part is trivial. So this paper concentrates on the \(\psi\)-part with \(\psi\) an even character. Theorem 2 says that if for every \(\alpha\) (in the \(\psi\)-part), the corresponding extension, if unramified, has a RNB, then the Iwasawa \(\Lambda\)-invariant vanishes. Theorem 3 gives a criterion in terms of a congruence modulo \(p\) of special values of \(L\)-functions. Theorem 4 is a kind of capitulation result: every unramified cyclic extension of degree \(p\) over \(k_n\) gets a RNB when passing over \(k_{n+1}\). Theorem 1 gives the application to the existence of a RNB when \(p\) does not divide the + part of the class number.
scientific article; zbMATH DE number 1050410