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Sufficient condition for polynomial irreducibility
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it.

Eisenstein's criterion may apply either directly (i. e., using the original polynomial) or after transformation of the original polynomial.

Consider the polynomial 1=Q(x) = 3x^(4) + 15x^(2) + 10. In order for Eisenstein's criterion to apply for a prime number p it must divide both non leading coefficients 15 and 10, which means only 1=p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10. One may therefore conclude that Q is irreducible over 'Q' (and, since it is primitive, over 'Z' as well). Note that since Q is of degree 4, this conclusion could not have been established by only checking that Q has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible.

Often Eisenstein's criterion does not apply for any prime number. It may however be that it applies (for some prime number) to the polynomial obtained after substitution (for some integer a) of x + a for x. The fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift. For example consider 1=H = x^(2) + x + 2, in which the coefficient 1 of x is not divisible by any prime, Eisenstein's criterion does not apply to H. But if one substitutes x for x + 3 in H, one obtains the polynomial x^(2) + 7x + 14, which satisfies Eisenstein's criterion for the prime number 7. Since the substitution is an automorphism of the ring 'Q'[x], the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally. In this particular example it would have been simpler to argue that H (being monic of degree 2) could only be reducible if it had an integer root, which it obviously does not; however the general principle of trying substitutions in order to make Eisenstein's criterion apply is a useful way to broaden its scope. Another possibility to transform a polynomial so as to satisfy the criterion, which may be combined with applying a shift, is reversing the order of its coefficients, provided its constant term is nonzero (without which it would be divisible by x anyway). This is so because such polynomials are reducible in R[x] if and only if they are reducible in R[x, x^(−1)] (for any integral domain R), and in that ring the substitution of x^(−1) for x reverses the order of the coefficients (in a manner symmetric about the constant coefficient, but a following shift in the exponent amounts to multiplication by a unit). As an example 2x^(5) − 4x^(2) − 3 satisfies the criterion for 1=p = 2 after reversing its coefficients, and (being primitive) is therefore irreducible in 'Z'[x].

An important class of polynomials whose irreducibility can be established using Eisenstein's criterion is that of the cyclotomic polynomials for prime numbers p. Such a polynomial is obtained by dividing the polynomial x^(p) − 1 by the linear factor x − 1, corresponding to its obvious root 1 (which is its only rational root if p > 2):

Here, as in the earlier example of H, the coefficients 1 prevent Eisenstein's criterion from applying directly. However the polynomial will satisfy the criterion for p after substitution of x + 1 for x: this gives

all of whose non leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient is equal to p, and therefore not divisible by p^(2). An alternative way to arrive at this conclusion is to use the identity 1=(a + b)^(p) = a^(p) + b^(p) which is valid in characteristic p (and which is based on the same properties of binomial coefficients, and gives rise to the Frobenius endomorphism), to compute the reduction modulo p of the quotient of polynomials:

which means that the non leading coefficients of the quotient are all divisible by p; the remaining verification that the constant term of the quotient is p can be done by substituting 1 (instead of x + 1) for x into the expanded form x^(p−1) + + x + 1.

The proof of this generalization is similar to the one for the original statement, considering the reduction of the coefficients modulo 'p'; the essential point is that a single term polynomial over the integral domain D/'p' 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).

After 'Z', one of the basic examples of an integral domain is the polynomial ring 1=D = k[u] in the variable u over the field k. In this case, the principal ideal generated by u is a prime ideal. Eisenstein's criterion can then be used to prove the irreducibility of a polynomial such as 1=Q(x) = x^(3) + ux + u in D[x]. Indeed, u does not divide a3, u^(2) does not divide a0, and u divides a0, a1 and a2. This shows that this polynomial satisfies the hypotheses of the generalization of Eisenstein's criterion for the prime ideal 1='p' = (u) since, for a principal ideal (u), being an element of (u) is equivalent to being divisible by u.