Eisenstein's Criterion - Generalization

Generalization

Given an integral domain D, let be an element of D, the polynomial ring with coefficients in D.

Suppose there exists a prime ideal of D such that

• a_i ∈ for each i ≠ n,
• a_n ∉ ,
• a₀ ∉ (where is the ideal product of with itself).

Then Q cannot be written as a product of two non-constant polynomials in D. If in addition Q is primitive (i.e., it has no non-trivial constant divisors), then it is irreducible in D. If D is a unique factorization domain with field of fractions F, then by Gauss's lemma Q is irreducible in F, whether or not it is primitive (since constant factors are invertible in F); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of D. The latter statement gives original theorem for D = Z or (in Eisenstein's formulation) for D = Z.

The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo ; the essential point is that a single-term polynomial over the integral domain cannot decompose as a product in which at least one of the factors has more than one term (because in such a product there can be no cancellation in the coefficient either of the highest or the lowest possible degree).