Bober, J. W. and Du, Lara and Fretwell, D. and Kopp, G. S. and Wooley, T. D. (2025) On 2-superirreducible polynomials over finite fields. Indagationes Mathematicae, 36 (3). pp. 753-763. ISSN 0019-3577
Full text not available from this repository.Abstract
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 F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in 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.