On the action of \(\text{GL}_2(\mathbb F_q)\) on irreducible polynomials over \(\mathbb F_q\)
type: Item
instance of: scholarly article
title: On the action of \(\text{GL}_2(\mathbb F_q)\) on irreducible polynomials over \(\mathbb F_q\)
zbMATH Open document ID: 1239.12003
author: Theodoulos Garefalakis
published in: Journal of Pure and Applied Algebra
publication date: 2011-05-10T00:00:00Z
review text: The author shows that \(\text{GL}_2(\mathbb F_q)\) acts on the set of irreducible polynomials over \(\mathbb F_q\) of degree at least 2. This action is studied and the cardinality of the set of fixed points of certain subgroups of \(\text{GL}_2(\mathbb F_q)\) is computed. The results include enumeration of irreducible polynomials that are invariant under the substitution \(X\rightarrow X + b\) and under the substitution \(X\rightarrow aX\). One of the main objectives is the characterization of the sets of fixed points of the action of single elements of \(\text{GL}_2(\mathbb F_q)\) of order \(r\), where \(r\) is a prime divisor of \(q\) or \(q-1\) (or equivalently of cyclic subgroups) on the set \(I_n\) of monic irreducibles of degree \(n\), and their cardinalities are computed. In fact it is shown that the set of fixed points of the action of any element of a \(p\)-Sylow subgroup of \(\text{GL}_2(\mathbb F_q)\), where \(p\) is the characteristic of \(F_q\), relates to irreducibles that are invariant under the substitution \(X\rightarrow X + b\). The number of even irreducible polynomials of given degree is obtained. The results are combined to provide enumeration formulas for the fixed point of upper triangular matrices.
reviewed by: Jebrel M. Habeb
Mathematics Subject Classification ID: 12E20
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 5886343
zbMATH Keywords: finite fields
zbMATH Keywords: irreducible polynomials
zbMATH Keywords: group action
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.10.015
OpenAlex ID: W1250112088
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DOI: 10.1016/J.JPAA.2010.10.015
description: scientific article; zbMATH DE number 5886343