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

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