1

I am looking to compile a list of tests for irreducibility / reducibility for polynomials over a field $\mathbb{F}$.

In the case $\mathbb{F} = \mathbb{Z}$ or $\mathbb{Q}$, Eisenstein's criterion and Gauss' lemma together give a sufficient condition for a polynomial over either of these fields to be irreducible.

Are there analogues, or different tests entirely, for other fields? Algebraically closed fields such as $\mathbb{C}$ are uninteresting, but what can we do (for example) over fields of characteristic $p$?

José Carlos Santos
  • 427,504
probablystuck
  • 616
  • 1
    You might want to look at this question and its answers: https://math.stackexchange.com/questions/2544570/strategy-for-irreducibility-testing-in-mathbbzx/2758003 – Sil Apr 29 '18 at 08:16

1 Answers1

1

A polynomial of degree 2 or 3 over $\mathbb{F}$ is reducible iff it has a root in $\mathbb{F}$, since any factorization would involve a factor of degree 1.

Adayah
  • 10,468