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.
The latter possibility can be avoided by first making Q primitive, by dividing it by the greatest common divisor of its coefficients (the content of Q).
This division does not change whether Q 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 p (on the contrary it could make the criterion hold for some prime, even if it did not before the division).
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 2x5 − 4x2 − 3 satisfies the criterion for p = 2 after reversing its coefficients, and (being primitive) is therefore irreducible in Z[x].
Here, as in the earlier example of H, the coefficients 1 prevent Eisenstein's criterion from applying directly.
An alternative way to arrive at this conclusion is to use the identity (a + b)p = ap + bp 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:
Theodor Schönemann was the first to publish a version of the criterion,[1] in 1846 in Crelle's Journal,[3] which reads in translation That (x − a)n + pF(x) will be irreducible to the modulus p2 when F(x) to the modulus p does not contain a factor x−a.This formulation already incorporates a shift to a in place of 0; the condition on F(x) means that F(a) is not divisible by p, and so pF(a) is divisible by p but not by p2.
As stated it is not entirely correct in that it makes no assumptions on the degree of the polynomial F(x), so that the polynomial considered need not be of the degree n that its expression suggests; the example x2 + p(x3 + 1) ≡ (x2 + p)(px + 1) mod p2, shows the conclusion is not valid without such hypothesis.
Assuming that the degree of F(x) does not exceed n, the criterion is correct however, and somewhat stronger than the formulation given above, since if (x − a)n + pF(x) is irreducible modulo p2, it certainly cannot decompose in Z[x] into non-constant factors.
The application for which Eisenstein developed his criterion was establishing the irreducibility of certain polynomials with coefficients in the Gaussian integers that arise in the study of the division of the lemniscate into pieces of equal arc-length.
Remarkably Schönemann and Eisenstein, once having formulated their respective criteria for irreducibility, both immediately apply it to give an elementary proof of the irreducibility of the cyclotomic polynomials for prime numbers, a result that Gauss had obtained in his Disquisitiones Arithmeticae with a much more complicated proof.
In fact, Eisenstein adds in a footnote that the only proof for this irreducibility known to him, other than that of Gauss, is one given by Kronecker in 1845.
This is all the more surprising given the fact that two pages further, Eisenstein actually refers (for a different matter) to the first part of Schönemann's article.
In a note ("Notiz") that appeared in the following issue of the Journal,[5] Schönemann points this out to Eisenstein, and indicates that the latter's method is not essentially different from the one he used in the second proof.
From Gauss' lemma it follows that Q is reducible in Z[x] as well, and in fact can be written as the product Q = GH of two non-constant polynomials G, H (in case Q is not primitive, one applies the lemma to the primitive polynomial Q/c (where the integer c is the content of Q) to obtain a decomposition for it, and multiplies c into one of the factors to obtain a decomposition for Q).
But then necessarily the reductions modulo p of G and H also make all non-leading terms vanish (and cannot make their leading terms vanish), since no other decompositions of axn are possible in (Z/pZ)[x], which is a unique factorization domain.
A second proof of Eisenstein's criterion also starts with the assumption that the polynomial Q(x) is reducible.
Applying the theory of the Newton polygon for the p-adic number field, for an Eisenstein polynomial, we are supposed to take the lower convex envelope of the points where vi is the p-adic valuation of ai (i.e. the highest power of p dividing it).
Now the data we are given on the vi for 0 < i < n, namely that they are at least one, is just what we need to conclude that the lower convex envelope is exactly the single line segment from (0, 1) to (n, 0), the slope being −1/n.
This tells us that each root of Q has p-adic valuation 1/n and hence that Q is irreducible over the p-adic field (since, for instance, no product of any proper subset of the roots has integer valuation); and a fortiori over the rational number field.
This argument is much more complicated than the direct argument by reduction mod p. It does however allow one to see, in terms of algebraic number theory, how frequently Eisenstein's criterion might apply, after some change of variable; and so limit severely the possible choices of p with respect to which the polynomial could have an Eisenstein translate (that is, become Eisenstein after an additive change of variables as in the case of the p-th cyclotomic polynomial).
In fact only primes p ramifying in the extension of Q generated by a root of Q have any chance of working.
For example, in the case x2 + x + 2 given above, the discriminant is −7 so that 7 is the only prime that has a chance of making it satisfy the criterion.
Again, for the cyclotomic polynomial, it becomes the discriminant can be shown to be (up to sign) pp−2, by linear algebra methods.
In fact, Eisenstein polynomials are directly linked to totally ramified primes, as follows: if a field extension of the rationals is generated by the root of a polynomial that is Eisenstein at p then p is totally ramified in the extension, and conversely if p is totally ramified in a number field then the field is generated by the root of an Eisenstein polynomial at p.[6] Given an integral domain D, let
If D is a unique factorization domain with field of fractions F, then by Gauss's lemma Q is irreducible in F[x], whether or not it is primitive (since constant factors are invertible in F[x]); 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[i].
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).