Construction of some wreath products as Galois groups of normal real extensions of the rationals
type: Item
instance of: scholarly article
title: Construction of some wreath products as Galois groups of normal real extensions of the rationals
zbMATH Open document ID: 0598.12010
DOI: 10.1016/0022-314X(86)90039-9
author: B. George
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(K\) be a real number field. Let \(f(X)\in K[X]\) be a real irreducible polynomial of degree \(n\) with Galois group \(G\) over \(K\). The author proves that there exists a rational number \(c\) such that the polynomial \(g(X)=f(X^ 2+c)\) is real with Galois group over \(K\) isomorphic to the wreath product of \({\mathbb Z}/2{\mathbb Z}\) by \(G\) of order {\#}G\(\cdot 2^ n\). Using this result he proves that a Sylow 2-subgroup of \(S_{2^ n}\), of \(A_{2^ n}\), and of \(\text{GL}(n,2)\), as well as the Weyl groups of type \(A_ n\), \(B_ n\) and \(D_ n\), occurs as a Galois group of a real normal extension of \({\mathbb Q}\). Remark that since this wreath product of \(G\) is a split extension of \(G\) with abelian kernel, it is known that it appears as a Galois group over \({\mathbb Q}\) [cf. \textit{J. Neukirch}, Invent. Math. 21, 59--116 (1973; Zbl 0267.12005)].
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20F29
zbMATH DE Number: 3963972
zbMATH Keywords: Galois group isomorphic to wreath product
zbMATH Keywords: inverse problem of Galois theory
zbMATH Keywords: Weyl groups
zbMATH Keywords: split extension
reviewed by: Nuria Vila
MaRDI profile type: Publication
cites work: A generalized group algebra for compact groups
cites work: Q5328221
cites work: Q4063245
cites work: Construction of fields of algebraic numbers with given solvable Galois group
cites work: On the Prime Divisors of the Sequence W n +1 = 1 + W 1 …W n
full work available at URL: 0022-314x(86)90039-9
OpenAlex ID: W2080880513
description: scientific article; zbMATH DE number 3963972