The number of irreducible polynomials with the first two prescribed coefficients over a finite field

Abstract

We use elementary combinatorial methods, together with the theory of quadratic forms, over finite fields to obtain the formula, originally due to Kuz'min, for the number of monic irreducible polynomials of degree~$n$ over a finite field $\mathbb{F} _q$ with the first two prescribed coefficients. The formula relates the number of such irreducible polynomials to the number of polynomials that split over the base field.