Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first. (English)

Normat. 57, No. 2, 49-73 (2009).

Summary: The Eisenstein criterion is well-known to all students of algebra, and it gives a very simple proof of the irreducibility of the cyclotomic polynomial $Θ_p(x)=x^{p-1}+x^{p-2}+\cdots+1$ for $p$ prime. The irreducibility was (of course) known to Gauss but via a much more complicated proof. Simple ideas almost always arrive through tortuous detours, and the Eisenstein criterion being no exception, was originally stated for Gaussian integers in connection with polynomials associated to the division problem on the lemniscate! When it was published in Crelle in 1850, it provoked a complaint from a now forgotten mathematician Schönemann (1812‒68) who had published a different proof of what was essentially the same criterion in the same journal only a few years earlier. His formulation is maybe even more elegant. Assume that $f(x)\in{\bbfZ}[x]$ is of degree $n>0$ and that there is some prime $p$ and an integer $a$ such that $f(x)=(x-a)^n+pF(x)$. If $F(a)\ne 0(p)$ then $f(x)$ irreducible mod $p^2$. This can be applied to the well known fact that $x^p-1=(x-1)^p(p)$ for $p$ prime to get the irreducibility of $Θ_p(x)$. Schönemann not only deserves the priority for the Eisenstein criterion (and for some time his name was actually attached to Eisenstein’s) but he also anticipated Hensel’s lemma (which, however, in some weaker form was later found in the Nachlaß of Gauss), and did work out a theory for finite fields, although of course scooped by both Gauss and Galois. Yet the modern presentation of the latter subject did not appear until the very end of the 19th century when lectured on by E. H. Moore. In addition to the main story a survey of the remarkable achievements of Gauss and Abel on elliptic functions is presented as providing the backdrop to the work of Schönemann and Eisenstein. Incidentally, the modern notion of an Abelian group, abstracted from the complicated equations on division points studied by Abel, did not appear until 1896 in Weber’s classical Lehrbuch der Algebra.