The Galois number
type: Item
instance of: scholarly article
title: The Galois number
zbMATH Open document ID: 0885.11027
author: Pat W. Beaulieu
author: Tommy C. Palfrey
published in: Mathematische Annalen
publication date: 1997-09-09T00:00:00Z
review text: Let \(G\) be a finite group acting on a nonempty finite set \(X\). The Galois number \(t_G\) of this action is defined to be \(1+M\), where \(M\) denotes the maximum among all numbers which occur as the number of fixed points of those elements in \(G\) which do not act as the identity. If \(f\) is an irreducible separable polynomial over a field \(K\) of degree \(n\) and \(N\) is the normal closure of \(K(\theta )\) where \(\theta\) is a root of \(f\), then \(G=\text{Gal}(N/K)\) acts on the roots of \(f\) and the Galois number of this action is just the smallest \(j\) such that any \(j\) roots of \(f\) generate \(N\) over \(K\) (cf. Corollary 2 in the present paper). The authors study certain identities in the Burnside ring \(B(G)\) that involve the above defined Galois numbers. In the situation \(G=\text{Gal}(N/K)\), where \(\text{char}(K)\neq 2\) and \(N\) is the normal closure of a degree \(n\) extension \(L/K\), they use these identities and a homomorphism from \(B(G)\) into the Witt ring \(W(K)\) to obtain vanishing theorems for the trace form \(\langle L\rangle\) of the extension \(L/K\). (The construction of this homomorphism is explained in detail.) For instance, they obtain the Witt ring identity \[ \prod_{m=0}^{t-1}(\langle L\rangle -m\cdot\langle 1\rangle )= \kappa_G\cdot\langle N\rangle, \] where \(t=t_G\) as above and \(\kappa_G= n!/((n-t)! [N:K])\) (Theorem 2), from which they deduce that the polynomial \((X-n\cdot 1)\prod (X-m\cdot 1)\), \(m\) ranging over the integers between \(0\) and \(t-1\) with \(m\equiv n\bmod 2\), vanishes on \(\langle L\rangle\). Except for the case \(t=n-1\), this polynomial is of lower degree than other general annihilating polynomials of trace forms of degree \(n\) that appeared previously in the literature [see \textit{J. Hurrelbrink}, Can. Math. Bull. 32, No. 4, 412-416 (1989; Zbl 0636.10019)]. The authors conclude with a list of examples where they evaluate \(t_G\) and \(\kappa_G\) for certain finite groups and certain a…
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 1058819
zbMATH Keywords: Galois number
zbMATH Keywords: group actions
zbMATH Keywords: Burnside ring
zbMATH Keywords: Witt ring
zbMATH Keywords: trace forms
zbMATH Keywords: annihilating polynomials of trace forms
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: s002080050103
OpenAlex ID: W2032430156
DOI: 10.1007/S002080050103
description: scientific article; zbMATH DE number 1058819