On Hasse principle for \(x^n=a\)
type: Item
instance of: scholarly article
title: On Hasse principle for \(x^n=a\)
zbMATH Open document ID: 0936.11060
author: Takashi Ono
author: Tomohide Terasoma
published in: Proceedings of the Japan Academy. Series A
publication date: 1998-09-06T00:00:00Z
review text: Let \(k\) be a, number field, let \(a\in k^*\) and let \(n>1\) be an integer. In this note it is proved that the equation \(x^n= a\) has a solution in \(k\) if and only if it has a solution in \(k_v\), for every place \(v\) of \(k\). This is called the Hasse principle for \(x^n= a\). As the authors remark, this is a special case of a more general theorem given by \textit{E. Artin} and \textit{J. Tate} [Class field theory, New York, Benjamin (1968; Zbl 0176.33504; see also Zbl 0681.12003)]. The proof given here uses properties of the Galois group of the splitting field of \(x^n=a\) and Chebotarev's density theorem. A consequence is that two elliptic curves \(E/k\) and \(E'/k\) are isomorphic over \(k\) if and only if they are isomorphic over every completion \(k_v\).
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 1160949
zbMATH Keywords: Hasse principle
zbMATH Keywords: Galois group
zbMATH Keywords: Chebotarev's density theorem
zbMATH Keywords: elliptic curves
reviewed by: Nuria Vila
MaRDI profile type: Publication
full work available at URL: pjaa.73.143
OpenAlex ID: W2070576753
cites work: Q5566638
cites work: Q3710637
DOI: 10.3792/PJAA.73.143
description: scientific article; zbMATH DE number 1160949