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Given a polynomial over a field, what are the methods to see it is irreducible? Only two comes to my mind now. First is Eisenstein criterion. Another is that if a polynomial is irreducible mod p then it is irreducible. Are there any others?

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    Well, Eisenstein's criterion is basically a version of reducing modulo a prime.2010-08-09
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    See also this meta answer for grouped list of questions related to this topic https://math.meta.stackexchange.com/questions/1868/list-of-generalizations-of-common-questions/28876#28876.2018-08-11

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