Cyclotomic extensions of number fields
type: Item
instance of: scholarly article
title: Cyclotomic extensions of number fields
zbMATH Open document ID: 1056.11058
DOI: 10.1016/S0019-3577(03)90003-6
author: Henri Cohen
author: Francisco Diaz y Diaz
author: Michel Olivier
published in: Indagationes Mathematicae. New Series
publication date: 2004-06-15T00:00:00Z
review text: Let \(K\) be a number field, \(l\) a prime number, and \(\zeta_l\) a primitive \(l\)th root of unity. The paper is devoted to the cyclotomic extension \(K(\zeta_l)/K\), giving explicit formulae for the discriminant, conductor, and different of this extension. Fixing a prime ideal \(\mathfrak p\) of \(K\) above \(l\), let \(P\) be the product of all prime ideals of \(K(\zeta_l)\) above \(\mathfrak p\). For any \(0\leq a\leq b\) such that \(P^a/P^b\) is an \(\mathbb F_l\)-module (i.e., \(b-a\) is less than or equal to the absolute ramification index of a prime of \(K(\zeta_l)\) above \(\mathfrak p\)) the Galois module structure of \(P^a/P^b\) is obtained. For any field \(M\) such that \(K\subseteq M\subseteq K(\zeta_l)\), the ramification index and the residual degree of a prime of \(M\) above \(\mathfrak p\) are given explicitly. In addition, the Galois module structure of the unit group of \(K(\zeta_l)\) modulo \(l\)th powers is derived.
reviewed by: Radan Kučera
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 2075408
zbMATH Keywords: discriminant
zbMATH Keywords: conductor
zbMATH Keywords: different
zbMATH Keywords: prime ideal decomposition
zbMATH Keywords: Galois module structure of unit group
MaRDI profile type: Publication
cites work: Densité des discriminants des extensions cycliques de degré premier
cites work: On the density of discriminants of cyclic extensions of prime degree
cites work: Q5612629
description: scientific article; zbMATH DE number 2075408