Global unit squares and local unit squares
type: Item
instance of: scholarly article
title: Global unit squares and local unit squares
zbMATH Open document ID: 1236.11093
author: Yan Li
author: Xianke Zhang
published in: Journal of Number Theory
publication date: 2008-11-18T00:00:00Z
review text: Let \(K\) be any Galois extension of \(\mathbb Q\), and \(U_K\) be the unit group of \(K\). For any place \(v\) of \(K\), let \(U_v\) be the group of local units of \(K_v\). Here the authors study the following problem: does there exist an odd prime \(p\) such that the map \(U_K/U_K^2\to \prod_{v\mid p} U_v/U_v^2\) is injective. They prove that the existence of such primes is equivalent to \(\Hom(U_K/U_K^2, \{\pm 1\})\) is a cyclic \(\mathbb F_2[\text{Gal}(K/\mathbb Q)]\)-module. Moreover, they also prove that if the class number \(h_{\mathbb Q(\zeta_{p^r})^+}\) is odd, then such primes exist for \(\mathbb Q(\zeta_{p^r})^+\) and \(\mathbb Q(\zeta_{p^r})\), where \(p\) is an odd prime and \(\mathbb Q(\zeta_{p^r})^+\) is the maximal real subfield of \(\mathbb Q(\zeta_{p^r})\).
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 5368238
zbMATH Keywords: unit square
zbMATH Keywords: Dirichlet density
zbMATH Keywords: Artin symbol
zbMATH Keywords: signature map
zbMATH Keywords: cyclotomic units
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.03.007
OpenAlex ID: W2026682149
cites work: Units with norm - 1 and signatures of units.
cites work: Q5843509
cites work: Square classes of totally positive units
cites work: Q4254917
cites work: Q5596313
cites work: Q5690489
DOI: 10.1016/J.JNT.2008.03.007
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description: scientific article; zbMATH DE number 5368238