Periods of maps on irreducible polynomials over finite fields
type: Item
instance of: scholarly article
title: Periods of maps on irreducible polynomials over finite fields
zbMATH Open document ID: 0912.11051
author: Patrick Morton
published in: Finite Fields and their Applications
publication date: 1999-03-29T00:00:00Z
review text: A polynomial \(\sigma(x)\) over a field \(F\) defines a dynamical system on the algebraic closure of \(F\). It is expedient to study the action of \(\sigma(x)\) on irreducible polynomials rather than on field elements: \(\sigma(x)\) maps an irreducible polynomial \(g(x)\) over \(F\) to the minimal polynomial of \(\sigma(\alpha)\), where \(\alpha\) is a root of \(g(x)\). In the case where \(F\) is the finite field \(F_q\), and \(\sigma(x)=x^q+ax\) (\(a\) is a non-zero element of \(F_q\)), the author proves that for every positive integer \(n\) there are infinitely many polynomial periodic orbits of minimal period \(n\). The proof highlights the dynamical significance of the so-called Carlitz module.
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 12E20
zbMATH DE Number: 1199984
zbMATH Keywords: algebraic dynamics
zbMATH Keywords: periodic polynomials
zbMATH Keywords: Carlitz module
reviewed by: Franco Vivaldi
MaRDI profile type: Publication
full work available at URL: ffta.1996.0168
OpenAlex ID: W2052824821
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DOI: 10.1006/FFTA.1996.0168
description: scientific article; zbMATH DE number 1199984