Effective criteria for \(p\)-th powers in a Galois extension
type: Item
instance of: scholarly article
title: Effective criteria for \(p\)-th powers in a Galois extension
zbMATH Open document ID: 0892.11039
author: Georges Gras
author: Stéphane Jeannin
published in: Journal of Number Theory
publication date: 1998-07-13T00:00:00Z
review text: This paper addresses the problem of testing when an algebraic integer \(e\) in a number field \(K\) is a \(p\)-th power (with \(p\) a prime number). The second author gave a criterion for that in his thesis. The paper begins by recalling that criterion. After that it is applied to the \(\chi\)-objects (with \(\chi\) a Galois character corresponding to \(K\)). The main part is section 4, which introduces a trace criterion: given a numerical approximation \(\epsilon_\sigma\) of the conjugate \(\sigma(e)\), the paper gives a device allowing to affirm that \(e\) is in \({K^*}^p\): a weighted sum \(\sum\zeta_\sigma \epsilon_\sigma\) should be near enough from a rational integer: this can be turned easily into a PARI program.
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 1011058
zbMATH Keywords: algebraic integer
zbMATH Keywords: Galois extension
zbMATH Keywords: Galois character
zbMATH Keywords: trace criterion
reviewed by: Roland Gillard
MaRDI profile type: Publication
full work available at URL: jnth.1997.2085
OpenAlex ID: W1551519959
cites work: Q4132630
cites work: Méthodes et algorithmes pour le calcul numérique du nombre de classes et de unités des extensions cubiques cycliques de Q.
cites work: Q3889159
cites work: Class numbers and units of E. Lehmer's cyclic quintic fields
cites work: Q5454198
DOI: 10.1006/JNTH.1997.2085
description: scientific article; zbMATH DE number 1011058