Counting irreducible polynomials over finite fields using the inclusion-exclusion principle

TL;DR

This paper presents a straightforward derivation of Gauss's formula for counting irreducible polynomials over finite fields using elementary field theory and the inclusion-exclusion principle.

Contribution

It offers an intuitive and immediate understanding of Gauss's formula through basic mathematical principles, simplifying its derivation.

Findings

Derivation of Gauss's formula using elementary methods

Insight into the structure of irreducible polynomials over finite fields

Simplification of the proof process for counting irreducible polynomials

Abstract

C. F. Gauss discovered a beautiful formula for the number of irreducible polynomials of a given degree over a finite field. Assuming just a few elementary facts in field theory and the exclusion-inclusion formula, we show how one see the shape of this formula and its proof instantly.