Eisenstein's Criterion

In mathematics, Eisenstein's criterion gives an easily checked sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorable into the product of non constant polynomials with rational coefficients. The result is also known as the Schönemann–Eisenstein theorem; although this name is rarely used nowadays, it was common in the early 20th century.

Suppose we have the following polynomial with integer coefficients.

If there exists a prime number such that the following three conditions all apply:

• divides each for ,
• does not divide, and
• does not divide ,

then is irreducible over the rational numbers. It will also be irreducible over the integers, unless all its coefficients have a nontrivial factor in common (in which case as integer polynomial will have some prime number, necessarily distinct from, as an irreducible factor). The latter possibility can be avoided by first making primitive, by dividing it by the greatest common divisor of its coefficients (the content of ). This division does not change whether is reducible or not over the rational numbers (see Primitive part–content factorization for details), and will not invalidate the hypotheses of the criterion for (on the contrary it could make the criterion hold for some prime, even if it did not before the division).

This criterion is certainly not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but does allow in certain important particular cases to prove irreducibility with very little effort. In some cases the criterion does not apply directly (for any prime number), but it does apply after transformation of the polynomial, in such a way that irreducibility of the original polynomial can be concluded.