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.
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Communicated by D. Panario.
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Cao, X., Hu, L. On the reducibility of some composite polynomials over finite fields. Des. Codes Cryptogr. 64, 229–239 (2012). https://doi.org/10.1007/s10623-011-9566-1
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DOI: https://doi.org/10.1007/s10623-011-9566-1