The quadratic and quartic character of certain quadratic units. I
type: Item
instance of: scholarly article
title: The quadratic and quartic character of certain quadratic units. I
zbMATH Open document ID: 0335.12006
author: Philip A. Leonard
author: Kenneth S. Williams
published in: Pacific Journal of Mathematics
publication date: 1977-01-01T00:00:00Z
review text: Let \(\varepsilon_n\) denote the fundamental unit of the real quadratic field \(\mathbb Q(\sqrt m)\). Interpreting \(\sqrt m\) as a rational integer modulo \(p\), the authors study the relationship between (rational) quadratic and biquadratic residue characters of \(\varepsilon_n\) modulo \(p\), and representations of \(p\) by positive-definite binary quadratic forms. Following an idea of \textit{P. Barrucand} and \textit{H. Cohn} [J. Reine Angew. Math. 238, 67--70 (1969; Zbl 0207.36202)], they obtain results related to the 21 imaginary bicyclic biquadratic fields which have class number 1 and which contain one of \(\sqrt{-1}\), \(\sqrt 2\) or if \(\sqrt{-2}\). In addition to giving new proofs of several known results, the authors prove some conjectures of \textit{Emma Lehmer} [J. Reine Angew. Math. 268/269, 294--301 (1974; Zbl 0289.12007)] and some additional new results. Further results obtained by similar methods are the subject of a second paper with the same title [Rocky Mt. J. Math. 9, 683--692 (1979; Zbl 0386.12002)].
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11E16
zbMATH DE Number: 3523700
zbMATH Keywords: fundamental unit of real quadratic field
zbMATH Keywords: quadratic and biquadratic residue characters
zbMATH Keywords: representations of primes by positive-definite binary quadratic forms
MaRDI profile type: Publication
full work available at URL: pjm.1977.71.101
OpenAlex ID: W2076677594
DOI: 10.2140/PJM.1977.71.101
Recommended article: On the quartic character of quadratic units
Recommended article: On the quadratic character of quadratic units
Recommended article: Q4083786
Recommended article: Quartic residuacity and the quadratic character of certain quadratic irrationalities
Recommended article: Q4099716
Recommended article: On the quartic character of certain quadratic units and the representation of primes by binary quadratic forms
Recommended article: Characterizations of Quadratic Polynomials
Recommended article: Q3222910
Recommended article: A Characterization of the Quadratic Irrationals
Recommended article: The Quadratic Character of 2
description: scientific article; zbMATH DE number 3523700