Square-free smooth polynomials in residue classes and generators of irreducible polynomials

Abstract

Building upon the work of A. Booker and C. Pomerance (2017), we prove that for a prime power $q\ge 7$, every residue class modulo an irreducible polynomial $F\in {\mathbb{F}}_q[X]$ has a non-constant, square-free representative which has no irreducible factors of degree exceeding $\text{deg}F-1$. We also give applications to generating sequences of irreducible polynomials.

Updated after discovering a typo in a cited paper