Polynomials over \({\mathbb{Q}}\) solving an embedding problem
type: Item
instance of: scholarly article
title: Polynomials over \({\mathbb{Q}}\) solving an embedding problem
zbMATH Open document ID: 0546.12006
author: Nuria Vila
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74678
review text: The fields defined by the polynomials constructed by \textit{E. Nart} and the author in [J. Number Theory 16, 6-13 (1983; Zbl 0511.12010), Th. 2.1], with absolute Galois group the alternating group \(A_ n\), can be embedded in any central extension of \(A_ n\) if and only if \(n\equiv 0 (mod 8)\), or \(n\equiv 2 (mod 8)\) and n is a sum of two squares. Consequently, for theses values of n, every central extension of \(A_ n\) occurs as a Galois group over \({\mathbb{Q}}\).
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 12E12
zbMATH DE Number: 3869466
zbMATH Keywords: embedding in central extension of \(A_ n\)
zbMATH Keywords: Galois group of equation
zbMATH Keywords: Galois extension with alternating group
zbMATH Keywords: central extension of \(A_ n\) as Galois group
zbMATH Keywords: polynomials
MaRDI profile type: Publication
OpenAlex ID: W2327167476
cites work: Endliche Gruppen I
cites work: Equations with absolute Galois group isomorphic to \(A_ n\).
cites work: The Witt invariant of the form \(\text{Tr}(x^ 2)\)
cites work: On central extensions of \(A_ n\) as Galois group over \({\mathbb{Q}}\)
DOI: 10.5802/AIF.1010
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 3869466