Some relations between the irreducible polynomials over a finite field and its quadratic extension

Abstract

In this paper, we establish some relations between irreducible polynomials over a finite field Fq and its quadratic extension Fq2. First we consider a relation between the numbers of irreducible polynomials of a fixed degree over Fq and Fq2, and some relations between self-reciprocal irreducible polynomials over Fq and self-conjugate-reciprocal irreducible polynomials over Fq2. We also obtain formulas for the number and the product of all self-conjugate-reciprocal irreducible monic (SCRIM) polynomials over Fq2.