96 Normat 57:2, 96 (2009)

Summary in English

David Cox, Why Eisenstein proved the

Eisenstein criterion and why Schöne-

mann discovered it ﬁrst.

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

+ · · · + 1 for p prime. The

irreducibility was (of course) known to

Gauss but via a much more compli-

cated proof. Simple ideas almost always

arrive through tortuous detours, and

the Eisenstein criterion being no excep-

tion, was originally stated for Gaussian

integers in connection with polynomi-

als associated to the division problem

on the lemniscate! When it was pub-

lished in Crelle in 1850, it provoked a

complaint from a now forgotten math-

ematician 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 el-

egant. Assume that f(x) ∈ Z[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) 6= 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 at-

tached to Eisenstein’s) but he also anti-

cipated Hensel’s lemma (which, how-

ever, in some weaker form was later

found in the Nachlaß of Gauss), and

did work out a theory for ﬁnite ﬁelds, al-

though of course scooped by both Gauss

and Galois. Yet the modern presenta-

tion 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 sur-

vey of the remarkable achievements of

Gauss and Abel on elliptic functions

is presented as providing the backdrop

to the work of Schönemann and Eisen-

stein. 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.

Lars Gårding, Näringskedjor

(Swedish).

A food-chain is an ordered collection of

populations feeding on each other. A

simple linear model for the variations of

the populations is introduced. A condi-

tion for stability, i.e. that the variations

of the populations are bounded, is that

all the eigenvalues of the corresponding

matrices lie on the unit-circle. The pop-

ulations will then vary cyclically. The

condition gives some restrictions on the

fertility and life-expectancies of the var-

ious populations. Special cases of sim-

ple chains are studied and compared

with real-life data, with some remark-

able good ﬁts.

Trond Steihaug and D.G. Rogers,

A minimum requiring angle trisection.

Given a piece of foldable material (e.g.

paper) in the shape of a right-angled

triangle. Fold it by placing the right-

angled corner on the hypothenuse. How

should this be done in order to mini-

mize the area of the folded triangle? The

problem leads to a cubic equation whose

relevant solution turns out to involve an

angle trisection (and thus obtainable by

a succession of foldings). Variations of

the problem are considered, in particu-

lar letting one of the acute corners in-

stead be placed on the opposite side.