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On a normal integral bases problem over cyclotomic \(\mathbb{Z}_p\)-extensions. (English) Zbl 0892.11036

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: 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.

MSC:

11R23 Iwasawa theory
11R18 Cyclotomic extensions
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers

Citations:

Zbl 0757.11044
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