Visibly irreducible polynomials over finite fields

TL;DR

This paper classifies certain polynomials over finite fields that can be proven irreducible using a specific structural approach involving sums of products, extending Lenstra's observations on cubic polynomials.

Contribution

It provides a classification of polynomials over finite fields that admit an irreducibility proof with a particular sum-of-products structure.

Findings

Identifies conditions under which polynomials are irreducible using this structure

Extends Lenstra's example to a broader class of polynomials

Provides a framework for analyzing polynomial irreducibility proofs

Abstract

H. Lenstra has pointed out that a cubic polynomial of the form (x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of {0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor divides one summand but not the other. We classify polynomials over finite fields that admit an irreducibility proof with this structure.