Cubic residues and binary quadratic forms
type: Item
instance of: scholarly article
title: Cubic residues and binary quadratic forms
zbMATH Open document ID: 1124.11006
published in: Journal of Number Theory
publication date: 2007-05-04T00:00:00Z
review text: One of the oldest results on higher reciprocity is Euler's observation that \(2\) is a cubic residue modulo a prime \(p \equiv 1 \bmod 3\) if and only if \(p\) is represented by the quadratic form \(x^2 + 27y^2\) of discriminant \(-27\cdot 2^2\). This conjecture was later proved by Gauss and generalized considerably by \textit{R. Dedekind} [J. Reine Angew. Math. 121, 40--123 (1899; JFM 30.0198.02), Ges. Math. Werke II, 340--353; (1900; JFM 30.0198.02)], who showed that for any integer \(m\) (not a cube) there is a subgroup \(H\) of index \(3\) in the class group of forms of discriminant \(-27m^2\) with the property that \(m\) is a cubic residue modulo a prime \(p \equiv 1 \bmod 3\) if and only if \(p\) is represented by a form in \(H\). Dedekind knew that \(m\) is a cubic residue modulo such a prime \(p\) if and only if \(p\) splits in the pure cubic extension \(\mathbb Q(\root 3\of {p})\), and conjectured that the result above generalizes to \textit{all} cubic fields. This conjecture was proved (and generalized from cubic to solvable extensions) by \textit{T. Takagi} [C. R. Acad. Sci. Paris 171, 1202--1205 (1920; JFM 47.0147.02)] using his newly developed class field theory; the part related to cubic extensions was rediscovered later by \textit{B. K. Spearman} and \textit{K. S. Williams} [J. Lond. Math. Soc. (2) 46, No. 3, 397--410 (1992; Zbl 0724.11002); ibid. 64, No. 2, 273--274 (2001; Zbl 1017.11017)]. Probably unaware of Dedekind's results and mainly motivated by Emma Lehmer's work on the residuacity of quadratic units, \textit{P. J. Weinberger} [Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder 1972, 241--242 (1972; Zbl 0321.12002)] showed that a unit in a quadratic number field with discriminant \(d\) is a cubic residue modulo a prime ideal \({\mathfrak p}\) if and only if the norm of \({\mathfrak p}\) is represented by a quadratic form in an explicitly given subgroup of index \(3\) in the class group of forms of discriminant \(-3df^2\), where \(f\) is …
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 5149371
zbMATH Keywords: cubic residues
zbMATH Keywords: cubic reciprocity
zbMATH Keywords: quadratic units
zbMATH Keywords: binary quadratic forms
author: Zhihong Sun
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.08.001
OpenAlex ID: W2122786434
cites work: Q3139838
cites work: Q4382830
cites work: Q3999066
cites work: On Euler's criterion
cites work: On the cubic character of quadratic units
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cites work: Q5813190
cites work: The Cubic Congruence x 3 + Ax 2 + Bx + C ≡ 0(mod p ) and Binary Quadratic Forms
cites work: THE CUBIC CONGRUENCE x3 + Ax2 + Bx + C ≡ 0 (mod p) AND BINARY QUADRATIC FORMS II
cites work: On the theory of cubic residues and nonresidues
cites work: Cubic and quartic congruences modulo a prime.
cites work: Quartic residues and binary quadratic forms
cites work: On the number of incongruent residues of \(x^4+ax^2+bx\) modulo \(p\)
cites work: Real quadratic fields with class numbers divisible by \(n\)
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DOI: 10.1016/J.JNT.2006.08.001
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description: scientific article; zbMATH DE number 5149371