On the norm groups of Galois \(2\mathfrak n\)-extensions of algebraic number fields
type: Item
instance of: scholarly article
title: On the norm groups of Galois \(2\mathfrak n\)-extensions of algebraic number fields
zbMATH Open document ID: 1176.11056
author: Leonid Stern
published in: Journal of Number Theory
publication date: 2009-05-19T00:00:00Z
review text: Let \(K\) be a finite extension of \({\mathbb Q}\), let \(L\) be a finite extension of \(K\), and let \(\overline {\mathbb Q}\) denote an algebraic closure of \({\mathbb Q}\). Let \(\text{Em}(L/K)\) denote the set of embeddings \(\tau: L\rightarrow \overline {\mathbb Q}\) which fix \(K\). The norm \(N_{L/K}: L\rightarrow K\) is defined as \(x\mapsto \prod_{\tau\in \text{Em}(L/K)} \tau(x)\), \(x\in L\). Let \(\{|\;|_\nu\}_{\nu\in W}\) be a set of representatives for the equivalence classes of normalized absolute values on \(L\). For each \(\nu\in W\), let \(L_\nu\) denote the completion of \(L\) with respect to the \(|\;|_\nu\)-topology on \(L\). For each discrete \(|\;|_\nu\), the ring of integers \(S_\nu\) of \(L_\nu\) is a compact subset of \(L_\nu\). Let \(\{Y_\nu\}_{\nu\in W}\) denote the family of subsets of \(L_\nu\) defined as \(Y_\nu=S_\nu\) if \(\nu\) is discrete and \(Y_\nu=L_\nu\), otherwise. The adele ring \(V_L\) of \(L\) is the ring \(\prod_{\nu\in W} L_\nu\) together with the restricted product topology on \(\prod_{\nu\in W} L_\nu\) with respect to the family \(\{Y_\nu\}\). There is an injection of commutative rings \(L\rightarrow V_L\). Let \((1_\nu)\) be defined as \(1_\nu=1\) for all \(\nu\in W\). Then \((1_\nu)\) is in \(V_L\), and so, \(V_L\) is a commutative ring with unity. The multiplicative group of units of \(V_L\) is the idele group of \(L\), denoted by \(I_L\). The norm map \(N_{L/K}\) can be defined on the \(I_L\) since for each \(\nu\in W\) lying above a prime \(Q\in K\), the set \((\text{Em}(L/K))_\nu=\{\tau\in \text{Em}(L/K):\;\tau(\nu)=\nu\}\) is a subset of \(\text{Em}(L/K)\). Note that \(\tau\in (\text{Em}(L/K))_\nu\) is an embedding \(\tau: L_\nu\rightarrow \overline {{\mathbb Q}_q}\) which fixes \(K_Q\). Here \(q\) is the rational prime which lies below \(Q\). In this manner \(N_{L/K}: I_L\rightarrow V_K\). The map \(C: I_L\rightarrow {\mathbb R}^+\) defined as \(C((a_\nu ))=\prod_{\nu\in W}| a_\nu|_\nu\) is a homomorphism of mult…
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 5554705
zbMATH Keywords: Galois theory
zbMATH Keywords: norm groups
zbMATH Keywords: class field theory
reviewed by: Robert G. Underwood
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.05.010
OpenAlex ID: W1999609074
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DOI: 10.1016/J.JNT.2008.05.010
description: scientific article; zbMATH DE number 5554705