id: 05701689
dt: j
an: 2010b.00049
au: Cox, David A.
ti: Why Eisenstein proved the Eisenstein criterion and why Schönemann
discovered it first.
so: Normat. 57, No. 2, 49-73 (2009).
py: 2009
pu: Nationellt Centrum för Matematikutbildning (NCM), Göteborgs Universitet,
Göteborg
la: EN
cc: A30 H20 H40
ut: history of mathematics; history of algebra; Eisenstein criterion;
irreducibility criterion; elliptic functions; lemniscate
ci:
li:
ab: 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.
rv: