Self-reciprocal irreducible polynomials with prescribed coefficients
type: Item
instance of: scholarly article
title: Self-reciprocal irreducible polynomials with prescribed coefficients
zbMATH Open document ID: 1246.11182
author: Theodoulos Garefalakis
published in: Finite Fields and their Applications
publication date: 2011-03-21T00:00:00Z
review text: Distribution of irreducible polynomials with additional properties over finite fields has attracted a lot of attention. This paper provides estimates for the number of self-reciprocal monic irreducible polynomials that having the \(t\) lower degree coefficients fixed to given values, over finite fields of odd characteristic. In particular, the existence of a self-reciprocal monic irreducible polynomial of degree \(2m\) with prescribed \(t\) low degree coefficients for \(t \leq m/2 - \log_q(2m) -1\) is established. The approach used in this paper is based on the work of \textit{L. Carlitz} [J. Reine Angew. Math. 227, 212--220 (1967; Zbl 0155.09801)], who has described monic self-reciprocal irreducible polynomials in terms of the form \(x^m P(x+x^{-1})\), where \(P\) is a monic irreducible polynomial of degree \(m\) such that \(x^2 -4\) is a non-square modulo \(P\), and also has enumerated of these polynomials using Dirichlet \(L\)-functions.
reviewed by: Qiang Wang
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 12E20
zbMATH DE Number: 5868431
zbMATH Keywords: self-reciprocal polynomials
zbMATH Keywords: irreducible polynomials
zbMATH Keywords: finite fields
MaRDI profile type: Publication
full work available at URL: j.ffa.2010.11.004
OpenAlex ID: W1965850767
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DOI: 10.1016/J.FFA.2010.11.004
description: scientific article; zbMATH DE number 5868431