On \(2\)-superirreducible polynomials over finite fields

We investigate \(k\)-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most \(k\). Let \(\mathbb F\) be a finite field of characteristic \(p\). We show that no \(2\)-superirreducible polynomials exist in \(\mathbb F[t]\) when \(p=2\) and that no such polynomials of odd degree exist when \(p\) is odd. We address the remaining case in which \(p\) is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree \(d\). This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.