The dynamics of permutations on irreducible polynomials

Abstract: 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 $\mathbb{F}q$ is induced by an action from a permutation polynomial of $\mathbb{F}{q^k}$ with coefficients in $\mathbb{F}_q$. The dynamics of these permutations of irreducible polynomials of degree $k$ over $\mathbb{F}_q$, 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.