Prove the following generalization of Eisenstein’s Criterion:
Let be a unique factorization domain with field of fractions and let be a prime ideal. If is a polynomial such that , for , and , then is irreducible in .
[Thanks to Marc van Leeuwen for suggesting this proof.]
We begin with a lemma.
Lemma: Let be an integral domain. If are polynomials such that and has only one nonzero term, then both and also have only one nonzero term. More specifically, and where and . Proof: Let and be the (nonzero) terms of highest and lowest degree in , and likewise let and be the nonzero terms of highest and lowest degree in . Now the term of highest degree in is and the term of lowest degree is . Thus we have , so that . Note that and are nonnegative, so that . That is, and , so that and each have only one nonzero term.
Now suppose is reducible in ; say , where and are nonunits. Reducing the coefficients mod , we have in . Because is prime, is an integral domain, and thus by the lemma both and have only one nonzero term. Suppose both or have positive degree. Then in particular, and (the constant terms of and ) are both in , so that , a contradiction. Thus without loss of generality, has degree 0, so that has degree 0. That is, where and is irreducible over .
Suppose now that is reducible in . Say . Clearing denominators, we have , where and are in . Then ; considering degrees, without loss of generality, is constant. But then is constant, and thus a unit in . Thus we have a contradiction, and so must be irreducible in .