Title: Irreducible compositions of degree two polynomials over finite fields have regular structure,
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.