Relative norms of Gauss sums
type: Item
instance of: scholarly article
title: Relative norms of Gauss sums
zbMATH Open document ID: 0608.12006
DOI: 10.1016/0022-314X(87)90038-2
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Consider the Gaussian sum \(S(\kappa)=\sum \kappa (u)e^{2\pi i tr(u)/p}\), where \(\kappa (u)\equiv u^{-(N{\mathfrak p}-1)/m}\) (mod \({\mathfrak p})\) is a fixed m-th power residue character modulo \({\mathfrak p}\), \({\mathfrak p}\) is an unramified prime in the m-th cyclotomic field \({\mathbb{Q}}(\zeta)\) lying above the odd rational prime p of residue degree f, \(tr(u)=u+u^ p+...+u^{p^{f-1}}\) and \(\zeta =e^{2\pi i/m}\). Now S(\(\kappa)\) lies in K(\(\zeta)\) for some abelian field K of conductor p and the author is interested in an exact calculation of \(N_{K(\zeta)/Kk}S(\kappa)\) where k is a subfield of \({\mathbb{Q}}(\zeta)\). He achieves this when m is odd and greater than 3 and k is a quadratic field, by making good use of Stickelberger's theorem and the analytic class number formula. The result is a nice application of this style of artihmetic.
reviewed by: John H. Loxton
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11L10
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 3983308
zbMATH Keywords: relative norms of Gauss sums
zbMATH Keywords: Gaussian sum
zbMATH Keywords: cyclotomic field
zbMATH Keywords: Stickelberger's theorem
zbMATH Keywords: analytic class number formula
author: Stanley J. Gurak
MaRDI profile type: Publication
cites work: Q5521595
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full work available at URL: 0022-314x(87)90038-2
OpenAlex ID: W2089887546
description: scientific article; zbMATH DE number 3983308