Counting irreducible polynomials over finite fields

Abstract

In this thesis we will look at a analogue of the prime number theorem for polynomials over a finite field. Using the formula of Gauss, we will derive an similar asymptotic formula for the number of monic irreducible polynomials with degree less or equal to a positive integer n. Unlike the prime number theorem, this result cannot be extended to the positive real numbers. In order to solve this issue, we will consider an encoding between polynomials over a finite field with q elements and the non-negative integers by writing the integers in base q. We consider the counting function that counts irreducible polynomials that are encoded by an integer smaller than a positive real number X. We then prove an analogue of the prime number theorem that does extend to the positive real numbers, by using a result by Pollack that grounded in Weil's Riemann Hypothesis for function fields.