Cubic fields, a congruential criterion for Scholz's theorem and new real quadratic fields with 3-rank equal to 4
type: Item
instance of: scholarly article
title: Cubic fields, a congruential criterion for Scholz's theorem and new real quadratic fields with 3-rank equal to 4
zbMATH Open document ID: 0642.12008
DOI: 10.1007/BF01190231
author: Francisco Diaz y Diaz
author: P. Llorente
author: Jordi Quer
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let k be a quadratic field of discriminant d and let \(H_ 3(d)\) be the subgroup of elements having order three of its ideal class group H(d). Let k be real; k' is defined as its associated imaginary quadratic field of discriminant \(d'=-3d/(3,d)\) 2. We denote by s and r the rank of the group \(H_ 3(d)\) and \(H_ 3(d')\), respectively. \textit{A. Scholz} [J. Reine Angew. Math. 166, 201-203 (1932; Zbl 0004.05104)] proves that: \(s\leq r\leq s+1.\) The authors compute s from the subgroup \(H_ 3(d')\) of order-three elements of H(d'). They give also the first known examples of quadratic fields with \(r=s=4\). He gives finally 40 non isomorphic totally real cubic fields K with discriminant \(d_ 0=314582172161\).
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 4045875
zbMATH Keywords: class number
zbMATH Keywords: quadratic fields
zbMATH Keywords: totally real cubic fields
zbMATH Keywords: discriminant
reviewed by: K. Lakkis
MaRDI profile type: Publication
cites work: On Some Families of Imaginary Quadratic Fields
cites work: Q4190711
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cites work: New types of quadratic fields having three invariants divisible by 3
cites work: Class Groups of the Quadratic Fields Found by F. Diaz y Diaz
cites work: A quadratic field of prime discriminant requiring three generators for its class group, and related theory
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description: scientific article; zbMATH DE number 4045875