# Three Proofs of an Observation on Irreducible Polynomials over $\text{GF}(2)$

## TL;DR

This paper provides three different proofs of an observation regarding the count of irreducible polynomials over GF(2) with specific properties, including one using a natural bijection, and also offers two proofs of a related observation.

## Contribution

It introduces multiple proofs, including a novel bijective approach, for counting irreducible polynomials over GF(2) with given trace and cotrace conditions.

## Key findings

- Three proofs of Ahmadi's observation on irreducible polynomials.
- Two proofs of a related polynomial count observation.
- One proof employs an explicit natural bijection.

## Abstract

We present three proofs of an observation of Ahmadi on the number of irreducible polynomials over $\text{GF}(2)$ with certain traces and cotraces, the most interesting of which uses an explicit natural bijection. We also present two proofs of a related observation.