Distinct Degree Factorizations for Polynomials over a Finite Field

Abstract

Let ${\widetilde {\mathbb {F}}_q}[X]$ denote the multiplicative semigroup of monic polynomials in one indeterminate $X$, over a finite field ${\mathbb {F}_q}$. We determine for each fixed $q$ and fixed $n$ the probability that a polynomial of degree $n$ in ${\mathbb {F}_q}[X]$ has irreducible factors of distinct degrees only. These results are of relevance to various polynomial factorization algorithms.