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.