On the computation of quadratic 2-class groups
type: Item
instance of: scholarly article
title: On the computation of quadratic 2-class groups
zbMATH Open document ID: 0870.11080
author: Wieb Bosma
author: Peter Stevenhagen
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1997-09-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/233222
full work available at URL: jtnb8-2.html
review text: The authors present a detailed description of an algorithm evolved from work of Gauss, D. Shanks, and \textit{J. C. Lagarias} [J. Algorithms 1, 142-186 (1980; Zbl 0473.68030)]. The idea is that the 2-class group for a discriminant \(D\) can be computed in cases where the full class group is not feasible, e.g., in random polynomial time, as long as the factors of \(D\) are known. The illustrations involve a \(D\) as a product of 5 primes close to \(10^{100}\). The prime factors immediately give ambiguous binary forms. Forms are successively examined by residue character for identification as a square. Then such a form is represented as (say) \(k^2\) (using ternary techniques), and the desired ``square-root-form'' is deduced from \((k^2,l,m)= 2(k,l,km)\). This is the detailed and time consuming part.
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11E20
zbMATH DE Number: 1002061
zbMATH Keywords: computation
zbMATH Keywords: ternary quadratic forms
zbMATH Keywords: 2-class group
zbMATH Keywords: ambiguous binary forms
reviewed by: Harvey Cohn
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W2334411759
cites work: The Magma algebra system. I: The user language
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cites work: Corrigenda
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DOI: 10.5802/JTNB.170
description: scientific article; zbMATH DE number 1002061