We study degree preserving maps over the set of irreducible polynomials over a finite field. In particular, we show that every permutation of the set of irreducible polynomials of degree k over Fq is induced by an action from a permutation polynomial of Fqk with coefficients in Fq. The dynamics of these permutations of irreducible polynomials of degree k over Fq, such as fixed points and cycle lengths, are studied. As an application, we also generate irreducible polynomials of the same degree by an iterative method.

Dynamics of finite fields, Fixed points, Irreducible polynomials, Permutation polynomials
Finite Fields and their Applications
School of Mathematics and Statistics

Reis, L. (Lucas), & Wang, Q. (2020). The dynamics of permutations on irreducible polynomials. Finite Fields and their Applications, 64. doi:10.1016/j.ffa.2020.101664