On irreducible polynomials of certain types in finite fields

Abstract

Let GF ( q ) be the finite field containing q = p l elements, where p is a prime and l a positive integer. Let P(x) be a monic polynomial in GF [ q, x ] of degree m . In this paper we investigate the nature and distribution of monic irreducible polynomials of the following types: (I) P ( x r ), where r is a positive integer ( r-polynomials ). (II) x m P ( x + x −1 ). ( Reciprocal polynomials .) These have the form (III) x rm P ( x r + x −r ). ( r-reciprocal polynomials .) These have the form Q ( x r ), where q ( x ) satisfies (1·1).