# Irreducible compositions of degree two polynomials over finite fields have regular structure

## TL;DR

This paper proves that the set of monic irreducible degree two polynomial compositions over finite fields has a regular structure, demonstrated through the construction of a finite automaton recognizing this set.

## Contribution

The paper introduces a constructive method to show that the set of such polynomials forms a regular language recognized by a finite automaton.

## Key findings

- The set of irreducible degree two polynomial compositions over finite fields is regular.
- A finite automaton recognizing this set is explicitly constructed.
- The structure of these polynomial compositions is regular and predictable.

## Abstract

Let $q$ be an odd prime power and $D$ be the set of monic irreducible polynomials in $\mathbb F_q[x]$ which can be written as a composition of monic degree two polynomials. In this paper we prove that $D$ has a natural regular structure by showing that there exists a finite automaton having $D$ as accepted language. Our method is constructive.