Norm groups of global fields
type: Item
instance of: scholarly article
title: Norm groups of global fields
zbMATH Open document ID: 0822.11074
published in: Journal of Number Theory
publication date: 1995-03-02T00:00:00Z
review text: \textit{L. Stern} [J. Number Theory 32, 203-219 (1989; Zbl 0687.12006); ibid. 36, 108-126 (1990; Zbl 0718.11056)] has proved the following theorems: (i) Let \(K/k\) and \(L/k\) be finite Galois extensions of a global field \(k\). Then \(K=L\) if and only if \(N_{K/k} K^ \times= N_{L/k} L^ \times\). (ii) If \(K/k\) is a nontrivial finite separable extension of global fields, then \(k^ \times/ N_{K/k} K^ \times\) is infinite. In this note, the author shows that these two results follow easily from the Chebotarev density theorem. The method of proof is to state the results for the images of the norm map on divisors rather than the element norm.
reviewed by: Richard Massy
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 729083
zbMATH Keywords: norm groups
zbMATH Keywords: finite Galois extensions
zbMATH Keywords: Chebotarev density theorem
zbMATH Keywords: norm map
author: Kevin Hutchinson
MaRDI profile type: Publication
full work available at URL: jnth.1995.1026
OpenAlex ID: W2059368854
DOI: 10.1006/JNTH.1995.1026
description: scientific article; zbMATH DE number 729083