On calculating the number \(N(D)\) of global cubic fields \(F\) of given discriminant \(D\)
type: Item
instance of: scholarly article
title: On calculating the number \(N(D)\) of global cubic fields \(F\) of given discriminant \(D\)
zbMATH Open document ID: 1493.11159
published in: Journal of Number Theory
publication date: 2022-03-18T00:00:00Z
review text: Let \(F\) be a global cubic field, i.e. a field extension \(F/F_0\) of degree \(3\), where \(F_0=\mathbb{Q}\) or \(F_0=\mathbb{F}_q(t)\). For given integral \(D\), i.e. \(D \in \mathbb{Z}\) or \(D \in \mathbb{F}_q [t]\), resp., let \(N(D)\) denote the number of non-isomorphic global fields \(F\) with discriminant \(D\). The method to determine \(N(D)\) uses class field theory and goes back to \textit{H. Hasse} [Math. Z. 31, 565--582 (1930); correction ibid. 31, 799 (1930; JFM 56.0167.02)] in the case of number fields and to the second author [Publ. Math. Debr. 79, No. 3--4, 611--621 (2011; Zbl 1249.11112)] in the case of function fields. The authors present a joint treatment for number fields and function fields, and first recall the results for Galois extensions \(F/F_0\). The calculation of \(N(D)\) in the non-Galois case is given in Theorem 3.3 and requires heavy computational methods. In Section 4 the authors present some new ideas and a new algorithm, which needs only calculations in the quadratic subfield of the Galois closure of \(F\). This makes it possible to obtain \(N(D)\) for much larger values of \(D\) than hitherto.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 7493035
zbMATH Keywords: discriminant
zbMATH Keywords: conductor
zbMATH Keywords: ray class group
zbMATH Keywords: Jacobi symbol
author: István Gaál
author: Michael E. Pohst
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.08.017
OpenAlex ID: W3208508757
Wikidata QID: Q114156744
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DOI: 10.1016/J.JNT.2021.08.017
description: scientific article; zbMATH DE number 7493035