Computation of integral bases in certain \(S_ n\) extensions of \({\mathbb{Q}}\)
type: Item
instance of: scholarly article
title: Computation of integral bases in certain \(S_ n\) extensions of \({\mathbb{Q}}\)
zbMATH Open document ID: 0634.12010
DOI: 10.1016/S0747-7171(87)80058-5
author: Eckart Maus
published in: Journal of Symbolic Computation
publication date: 1987-01-01T00:00:00Z
review text: Let L be the decomposition field of an irreducible polynomial \(f\in {\mathbb{Z}}[X]\) with Galois group \(S_ n\). Let K be the fixed quadratic field of \(A_ n\subset S_ n\). The author proves that discriminant \(disc(f)=disc(K)\) if and only if \({\mathbb{Z}}[\alpha_ 1,...,\alpha_ n]\) is the ring of integers in L and L/K is unramified, where \(\alpha_ 1,...,\alpha_ n\) are the roots of f in L. Examples for \(f=x^ n+ax^ m+b\) are given. The special case with disc(f) squarefree was also dealt with by \textit{J. Elstrodt}, \textit{F. Grunewald} and \textit{J. Mennicke} in Glasgow Math. J. 27, 31-37 (1985; Zbl 0579.12005).
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 12-04
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 4031823
zbMATH Keywords: computation of integral bases
zbMATH Keywords: decomposition field of an irreducible polynomial
zbMATH Keywords: discriminant
reviewed by: Zhang Xian
MaRDI profile type: Publication
cites work: On unramified Am-extensions of quadratic number fields
cites work: Zur Arithmetik einiger Serien nichtauflösbarer Gleichungen 5. Grades
cites work: Q4692759
cites work: Unramified extensions of quadratic number fields. II
cites work: On unramified Galois extensions of quadratic number fields
description: scientific article; zbMATH DE number 4031823