On irreducible polynomials of certain types in finite fields

Abstract

Let GF(q) be the finite field containing q = pl 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(xr), where r is a positive integer (r-polynomials).(II) xm P(x + x−1). (Reciprocal polynomials.) These have the form(III) xrmP(xr + x−r). (r-reciprocal polynomials.) These have the form Q(xr), where q(x) satisfies (1·1).