A construction of normal bases over the Hilbert \(p\)-class field of imaginary quadratic fields
type: Item
instance of: scholarly article
title: A construction of normal bases over the Hilbert \(p\)-class field of imaginary quadratic fields
zbMATH Open document ID: 0934.11054
author: Tsuyoshi Itoh
published in: Proceedings of the Japan Academy. Series A
publication date: 1998-11-05T00:00:00Z
review text: Soient \(p\) un nombre premier impair, \(F\) un corps quadratique imaginaire et \(K\) une \(\mathbb{Z}_p\)-extension de \(F\). Lorsque \(k\) est le corps des classes de rayon modulo \(p\) de \(F\), K. Komatsu établit l'existence d'une base normale pour la \(\mathbb{Z}_p\)-extension \(kK/k\). Dans cet article, l'auteur examine le cas où \(k\) désigne le corps des \(p\)-classes de Hilbert de \(F\). Plus précisément, il prouve que l'extension \(kK/k\) admet une base normale sauf dans le cas suivant: \[ p=3 \quad\text{et}\quad F=\mathbb{Q}(\sqrt{-3d}) \] avec \(d>1,\) \(d\equiv 1\pmod 3\) (\(d\) étant suppsé entier sans facteur carré). Les démonstrations reposent sur des lemmes de transposition d'une extension à une autre, du problème d'existence d'une base normale pour l'anneau des \(p\)-entiers.
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1192514
zbMATH Keywords: normal basis
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: ray class field
zbMATH Keywords: Hilbert class field
reviewed by: Florence Soriano-Gafiuk
MaRDI profile type: Publication
full work available at URL: pjaa.74.25
OpenAlex ID: W2088971461
DOI: 10.3792/PJAA.74.25
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description: scientific article; zbMATH DE number 1192514