Number of Irreducible Polynomials over Finite Fields with First Two Coefficients Fixed

Gary McGuire, Emrah Sercan Yılmaz

Published 2016-09-08Version 1

Let $p$ be an odd prime number. For any positive integers $n\geq 2, r\geq 1$ we present formulae for the number of irreducible polynomials of degree $n$ over the finite field $\mathbb F_{p^r}$ where the coefficients of $x^{n-1}$ and $x^{n-2}$ are fixed. Our proof involves counting the number of points on an algebraic curve over finite fields.