Finding a Parallel to Eisenstein's Criteria over Finite Fields

Abstract

Factoring polynomials over the integers is something that we all learn in high school algebra. Homework assignments are made up of countless polynomials and it is our job to find the factors. Eventually, the polynomials get longer and more difficult and we learn other methods to factor them. The problem becomes harder when you come across a polynomial which you are unable to factor. With certain polynomials like x^2+1, you can easily tell that it cannot be factored over Z. However, other times we are left wondering whether a given polynomial p(x) is factorable over Z at all. Lucky for us, Gotthold Eisenstein, a mathematician, found a sufficient condition for irreducibility over Z.