New relative extension examples without normal basis
type: Item
instance of: scholarly article
title: New relative extension examples without normal basis
zbMATH Open document ID: 1012.11099
author: Jean Cougnard
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2003-02-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73554
review text: The setup is the following: All fields are supposed abelian over \(\mathbb{Q}\); \(N\) is the disjoint compositum of two other fields \(K\) and \(k\), \(K\) is tame over \(\mathbb{Q}\) with Galois group \(H\) of odd prime order \(l\), and the question is whether \(N/k\) has NIB or at least WNIB, where NIB means ``normal integral basis'', i.e. \(O_N\) is free as an \(O_k[H]\)-module; and WNIB (weak NIB) means that \({\mathcal M}O_N\) is free over the maximal order \({\mathcal M}\) of \(O_k[H]\). In the case \(N={\mathbb{Q}}(\zeta_p)\) it was shown by \textit{J. Brinkhuis} [J. Reine Angw. Math. 375/376, 157-166 (1987; Zbl 0609.12009)] and by \textit{J. Cougnard} [J. Number Theory 23, 336-346 (1986; Zbl 0588.12003)] that \(N/k\) does not have NIB. (Note that the above assumptions imply that \(l\) divides \(p-1\) exactly once.) In the present paper, more choices are allowed for \(k\), with \(K\) still the degree \(l\) field of prime conductor \(p\), and it is proved that most of the time \(N/k\) does not even have WNIB. The exact conditions on \(k\) are: \(k\) is cyclic of degree \(n\) over the rationals; \(p\) is totally ramified in \(k\), and the non-\(p\)-part \(n_0\) of \(n\) is not divisible by \(l\). The exact result is: If then \(N/k\) has WNIB, then \(k\) is \(\mathbb{Q}(\sqrt{p})\) or \({\mathbb{Q}}(\sqrt{-p})\); and even in this case, there is no NIB. The reviewer [ibid. 35, 180-193 (1990; Zbl 0718.11053)] showed some 10 years ago that one has WNIB rather often in this special case, but the question whether this is always so remains undecided. The author puts the established methods (resolvents in particular) to very good use. One main point is the fact that in the minus part the classes of totally ramified prime ideals tend to generate a part of the class group that is as large as possible. There is a very cleverly contrived example on p. 499 where one can explicitly ``see'' the nonexistence of a WNIB. A few small comments: The initial of Brinkhuis in the abs…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1796111
zbMATH Keywords: normal integral bases
zbMATH Keywords: resolvents
zbMATH Keywords: Gauss sums
MaRDI profile type: Publication
OpenAlex ID: W2331057340
cites work: Galois modules and embedding problems.
cites work: Normal integral bases and complex conjugation.
cites work: Relative normal bases in some cyclotomic extensions
cites work: Arithmetic and Galois module structure for tame extensions.
cites work: Nombre de $\varphi$-classes invariantes. Application aux classes des corps abéliens
cites work: Relative integral normal bases in \({\mathbb{Q}}(\zeta_ p)\)
cites work: Q4177670
cites work: Introduction to Algebraic K-Theory. (AM-72)
DOI: 10.5802/AFST.999
description: scientific article; zbMATH DE number 1796111