Eisenstein's Criterion - History

History

The criterion is named after Gotthold Eisenstein. However, Theodor Schönemann was the first to publish a version of the criterion, in 1846 in Crelle's Journal, 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) modulo 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 into non-constant factors.

Subsequently Eisenstein published a somewhat different version in in 1850, also in Crelle's Journal. This version reads in translation

When in a polynomial F(x) in x of arbitrary degree the coefficient of the highest term is = 1, and all following coefficients are whole (real, complex) numbers, into which a certain (real resp. complex) prime number m divides, and when furthermore the last coefficient is = εm, where ε denotes a number not divisible by m: then it is impossible to bring F(x) into the form

(xμ + a₁xμ−1 + … + a_μ)(xν + b₁xν−1 + … + b_ν)

where μ and ν ≥ 1, μ + ν = the degree of F(x), and all a and b are whole (real resp. complex) numbers; the equation F(x) = 0 is therefore irreducible.

Here "whole real numbers" are ordinary integers and "whole complex numbers" are Gaussian integers; one should similarly interpret "real and complex prime numbers". 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 lemnsicate 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 shows that he was unaware of two different proofs of this statement that Schönemann had given, one in either part of a two-part article, the second of which being the one based on the criterion cited above; 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, 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.