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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 $p$?

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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.

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