Irreducibles and the composed product for polynomials over a finite field
type: Item
instance of: scholarly article
title: Irreducibles and the composed product for polynomials over a finite field
zbMATH Open document ID: 0615.05007
DOI: 10.1016/0012-365X(87)90135-X
author: Leonard Carlitz
author: Joel V. Brawley
published in: Discrete Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \(\mathrm{GF}(q)\) denote the finite field of \(q\) elements and let \(\mathrm{GF}[q,x]\) denote the integral domain of polynomials in an indeterminate \(x\) over \(\mathrm{GF}(q)\). Further, let \(\Gamma =\Gamma (q)\) denote the algebraic closure of \(\mathrm{GF}(q)\) so that every polynomial in \(\mathrm{GF}[q,x]\) factors completely in \(\Gamma\). This paper considers certain sets of monic polynomials from \(\mathrm{GF}[q,x]\) on which there is defined a binary operation called the composed product. Here, if \(f\) and \(g\) are monics in \(\mathrm{GF}[q,x]\) with \(\deg f=m\) and \(\deg g=n\), then the composed product, denoted by \(f\diamond g\) and defined in terms of the roots of \(f\) and \(g\), is also in \(\mathrm{GF}[q,x]\) and has degree \(mn\). In the present paper, the two most important composed products, denoted by the special symbols \(\circ\) and \(*\), are those induced by the field multiplication and the field addition on \(\Gamma\) and defined by: \[ f\circ g=\prod_{\alpha}\prod_{\beta}(x-\alpha \beta),\quad f*g=\prod_{\alpha}\prod_{\beta}(x-(\alpha +\beta)), \] where the products indicated by \(\prod\) are the usual products in \(\Gamma[x]\) and are taken over all the roots \(\alpha\) of \(f\) and \(\beta\) of \(g\), (including multiplicities). These two composed products are called composed multiplication and composed addition, respectively. After introducing and developing some theory concerning a more general notion of composed product, this paper moves to the special composed products above and asks whether the irreducibles over \(\mathrm{GF}(q)\) can be factored uniquely into indecomposables with respect to each of these products. Here, the term ``irreducible'' is used in the usual sense of the word while the term ``indecomposable'' is used in reference to composed products. This question is shown to have an affirmative answer in both situations, and thus yield unique factorization theorems (multiplicative and additive) for \(\Gamma\). …
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 3995690
zbMATH Keywords: monic polynomials
zbMATH Keywords: composed product
zbMATH Keywords: factorization theorems
zbMATH Keywords: inversion formula
MaRDI profile type: Publication
cites work: Q5514658
cites work: Q3216750
cites work: Q4754499
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
cites work: Q5831515
cites work: Polynomials and linear transformations over finite fields.
full work available at URL: 0012-365x(87)90135-x
OpenAlex ID: W2051677705
description: scientific article; zbMATH DE number 3995690