On the reducibility of some composite polynomials over finite fields

Abstract

Let g(x) = x n + a n-1 x n-1 + . . . + a 0 be an irreducible polynomial over $${\mathbb{F}_q}$$ . Varshamov proved that for a = 1 the composite polynomial g(x p ?ax?b) is irreducible over $${\mathbb{F}_q}$$ if and only if $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})\neq 0}$$ . In this paper, we explicitly determine the factorization of the composite polynomial for the case a = 1 and $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})= 0}$$ and for the case a ? 0, 1. A recursive construction of irreducible polynomials basing on this composition and a construction with the form $${g(x^{r^kp}-x^{r^k})}$$ are also presented. Moreover, Cohen's method of composing irreducible polynomials and linear fractions are considered, and we show a large number of irreducible polynomials can be obtained from a given irreducible polynomial of degree n provided that gcd(n, q 3 ? q) = 1.