Schonemann's criterion
If
- is monic
- , a prime and an integer such that
- is an irreducible polynomial in and does not divide
then is irreducible.
Proof
We know that is monic, so deg deg and that is monic. Assume , where . Since , we get , so . Therefore, we have and for some and . Therefore, This means that , which means that , a contradiction. This means that is irreducible.
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See also Eisenstein's criterion.