Irreducibles and the composed product for polynomials over a finite field
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 \(\Gamma\). These theorems are then used to prove corresponding unique factorization theorems for all subfields of \(\Gamma\). Next, it is shown that there are no irreducibles \(f\) in \(\mathrm{GF}[q,x]\) which can be decomposed as \(f=f_ 1\circ g_ 1=f_ 2*g_ 2\) (except for trivial decompositions). A special inversion formula is then derived and using this inversion formula, the authors determine the numbers of irreducibles of degree \(n\) which are indecomposable with respect to (i) composed multiplication \(\circ\), (ii) composed addition \(*\), and (iii) both the composed products \(\circ\) and \(*\) simultaneously. These numbers are given in terms of the well-known number of irreducibles of degree \(n\) over \(\mathrm{GF}(q)\). A final section contains some discussion and several observations about the more general composed product.
scientific article; zbMATH DE number 3995690