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

TL;DR

This paper proves that for prime powers q ≥ 7, every residue class modulo an irreducible polynomial over finite fields has a square-free polynomial representative with controlled degree factors, with applications to generating irreducible polynomials.

Contribution

It extends previous work by showing the existence of specific square-free polynomial representatives in residue classes over finite fields with degree constraints.

Findings

Existence of square-free polynomial representatives in all residue classes for q ≥ 7.

Construction of sequences of irreducible polynomials using these representatives.

Applications to polynomial generation algorithms in finite fields.

Abstract

Building upon the work of A. Booker and C. Pomerance (2017), we prove that for a prime power $q \geq 7$, 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.