Power residue character of rational primes
type: Item
instance of: scholarly article
zbMATH Open document ID: 1018.11003
author: Charles Helou
author: Ron Roll
author: Lawrence C. Washington
publication date: 2003-09-12T00:00:00Z
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 1690605
zbMATH Keywords: Eisenstein's reciprocity law
zbMATH Keywords: cubic residues
zbMATH Keywords: power residues
MaRDI profile type: Publication
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title: Power residue character of rational primes
published in: Journal of the Ramanujan Mathematical Society
review text: Let \(p\) and \(q\) be odd primes such that \(q \equiv 1 \bmod p\). It is a classical problem to characterize the primes \(r \neq q\) that are \(p\)th power residues modulo \(q\). Euler, for example, conjectured that \(2\) is a cubic residue modulo \(q\) if and only if \(q\) is represented by the quadratic form \(x^2 + 27y^2\). Criteria for fifth power residues involve systems of quadratic forms. In this article, the authors use Eisenstein's reciprocity law for rational integers and primary elements in the cyclotomic ring \(\mathbb Z[\zeta_p]\) to derive a general criterion, and then they give applications to cubic residuacity as well as to the \(p\)th power residue character of small primes for \(p = 5\), \(7\) and \(11\).
reviewed by: Franz Lemmermeyer
description: scientific article; zbMATH DE number 1690605