Normat 57:2, 49–73 (2009) 49

Why Eisenstein proved the Eisenstein criterion

and why Schönemann discovered it ﬁrst

David A. Cox

Department of Mathematics

Amherst College

Amherst, MA 01002, USA

dac@cs.amherst.edu

The Eisenstein irreducibility critierion is part of the training of every mathemati-

cian. I ﬁrst learned the criterion as an undergraduate and, like many before me,

was struck by its power and simplicity. This article will describe the unexpectedly

rich history of the discovery of the Eisenstein criterion and in particular the role

played by Theodor Schönemann.

For a statement of the criterion, we turn to Dorwart’s 1935 article “Irreducibility

of polynomials” in the American Mathematical Monthly [9]. As you might expect,

he begins with Eisenstein:

The earliest and probably best known irreducibility criterion is the

Schoenemann-Eisenstein theorem:

If, in the integral polynomial

+ a

n−1

+ ··· + a

all of the coeﬃcients except a

are divisible by a prime p, but a

is not

divisible by p

, then the polynomial is irreducible.

Here’s our ﬁrst surprise—Dorwart adds Schönemann’s name in front of Eisenstein’s.

He then gives a classic application:

An important application of this theorem is the proof of the irre-

ducibility of the so-called “cyclotomic polynomial”

f(x) =

− 1

x − 1

= x

p−1

+ x

p−2

+ ··· + 1,

where p is prime.

50 David A. Cox Normat 2/2009

If, instead of f(x), we consider f(x + 1), where

f(x + 1) =

(x + 1)

− 1

(x + 1) − 1

= x

p−1





p−2

+ ··· + p,

the theorem is seen to apply directly, and the irreducibility of f (x + 1)

implies the irreducibility of f(x).

The combination “Schönemann-Eisenstein” (often “Schoenemann-Eisenstein”) was

common in the early 20th century. An exception is Dorrie’s delightful book Triumph

der Mathematik, published in 1933 [8], where he states the “Satz von Schoenemann.”

Another exception is van der Waerden’s Moderne Algebra from 1930 [28], where we

ﬁnd the “Eisensteinscher Satz.”

Given the inﬂuence of van der Waerden’s book on succeeding generations of

textbook writers, we can see how Schönemann’s name got dropped. But how did

it get added in the ﬁrst place? Equally important, how did Eisenstein’s get added?

And why both names? To answer these questions, we need to explore some 19th

century number theory. This is a rich subject, so by necessity my treatment will

be far from complete. I will instead focus on speciﬁc highlights to trace the de-

velopment of these ideas. There will be numerous quotes (with translations when

necessary) to illustrate how mathematics was done at the time and what it looked

like. We begin with Gauss.

Gauss

Disquisitiones Arithmeticae [13], published in 1801, contains an amazing amount

of mathematics. In particular, Gauss proves that when p is prime, the cyclotomic

polynomial x

p−1

+···+x+1 is irreducible. His proof uses an explicit representation

of the roots and is not easy. However, he also uses the following general result that

relates irreducibility over Z to irreducibility over Q:

42.

Si coëﬃcientes A, B, C . . . . N; a, b, c . . . . n duarum functionum for-

mae

+ Ax

m−1

+ Bx

m−2

+ Cx

m−3

. . . .. + N . . . . . . . (P )

+ ax

µ−1

+ bx

µ−2

+ cx

µ−3

. . . .. + n . . . . . . . (Q)

omnes sunt rationales, neque vero omnes integri, productumque ex (P )

et (Q)

= x

m+µ

+ Ax

m+µ−1

+ Bx

m+µ−2

+ etc. + Z

omnes coëﬃcientes A, B . . . . Z integri esse nequeunt.

This edition included a reference to Schönemann that was dropped in the 1937 second edition.

Normat 2/2009 David A. Cox 51

42.

If the coeﬃcients A, B, C . . . . N ; a, b, c . . . . n of two functions of the

form

+ Ax

m−1

+ Bx

m−2

+ Cx

m−3

. . . .. + N . . . . . . . (P )

+ ax

µ−1

+ bx

µ−2

+ cx

µ−3

. . . .. + n . . . . . . . (Q)

are all rational and not all integers, and if the product of (P ) and (Q)

= x

m+µ

+ Ax

m+µ−1

+ Bx

m+µ−2

+ etc. + Z

then not all the coeﬃcients A, B . . . . Z can be integers.

This is what we now call Gauss’s Lemma. His proof is essentially the same that

appears in abstract algebra texts, though he states the result in the contrapositive

form and never uses the term “polynomial.” Gauss also doesn’t use the three dots

··· that are standard today.

Another major result of Disquisitiones is Gauss’s proof that x

−1 = 0 is solvable

by radicals. The modern approach to solvability by radicals allows the introduction

of arbitrary roots of unity, which implies that x

−1 = 0 is trivially solvable. Gauss

instead followed the inductive strategy pioneered by Lagrange, where one constructs

the roots recursively using polynomials of strictly smaller degree that are solvable

by radicals. In modern terms, this gives an explicit description of the intermediate

ﬁelds of the extension

Q ⊆ Q(e

2πi/p

)

when p is prime. This has degree p − 1 by the irreducibility of x

p−1

+ ··· + x + 1.

From here, Gauss obtains his wonderful result about dividing the circle into n equal

arcs by straightedge and compass.

The second paragraph of Section VII of Disquisitiones begins with a famous

passage:

Ceterum principia theoriae, quam exponere aggredimur, multo latius

patent, quam hic extenduntur. Namque non solum ad functiones cir-

culares, sed pari successu ad multas alias functiones transscendentes

applicari possunt, e.g. ad eas, quae ad integrali

√

(1−x

)

pendent,

praetereaque etiam ad varia congrueniarum genera: sed quoniam de il-

lis functionibus transscendentibus amplum opus peculiare paramus, de

congruentiis autem in continuatione disquitionum arithmeticarum co-

piose tractabitur, hoc loco solas functiones circulares considerare visum

est.

The principles of the theory we are going to explain actually extend

much farther than we will indicate. For they can be applied not only

to circular functions but just as well to other transcendental functions,

e.g. to those which depend on the integral

√

(1−x

)

and also to various

types of congruences. Since, however, we are preparing a large work on

those transcendental functions and since we will treat congruences at

52 David A. Cox Normat 2/2009

length in the continuation of these Disquisitiones, we have decided to

consider only circular functions here.

In this quote, the reference to circular functions is clear. But what about transcen-

dental functions that depend on the integral

√

(1−x

)

? Here, any 19th century

mathematician would immediately think of the lemniscate r

= cos 2θ, whose arc

length is 4

√

(1−x

)

. This integral and its relation to the lemniscate were discov-

ered by the Bernoulli brothers in the late 17th century and played a key role in

the development of elliptic integrals by Fagnano, Euler, and Legendre in the 18th

century. Gauss’s “large work” on these functions never appeared, though fragments

found after Gauss’s death contain some astonishing mathematics (see [3]).

The quote also mentions “various types of congruences” that will be discussed “in

the continuation of these Disquisitiones.” The published version of Disquisitiones

had seven sections, but Gauss drafted an eighth section, Disquisitiones generales de

congruentiis, that studied polynomial congruences f(x) ≡ 0 mod p, where f ∈ Z[x]

and p is prime (see pp. 212–242 of [15, Vol. II] or pp. 602–629 of the German version

of [13]). In modern terms, Gauss is studying the polynomial ring F

[x]. Here are

some of his results:

• The existence and uniqueness of factorization of polynomials modulo p.

• A determination of the number (n) of monic irreducible polynomials modulo

p. His result is

n(n) = p

−

abc

etc.

where the sum

(resp.

) is over all distinct prime factors (resp.

pairs of distinct prime factors) of n, and similarly for the remaining terms in

the formula.

Gauss also had a theory of ﬁnite ﬁelds, though his approach is not easy for the

modern reader because of his reluctance to introduce roots of polynomial congru-

ences. Here is what Gauss says about the congruence ξ ≡ 0 mod p, where ξ is a

polynomial with integer coeﬃcients:

. . . ad hinc hihil obstat, quominus ξ in factores duram, trium pluri-

umve dimensionum resolvi possit, unde radices quasi imaginariae illi

attribui possint. Revera, si simili licentia, quam recentiores mathematici

usurparunt, uti talesque quantitates imaginarias introducere voluisse-

mus, omnes nostras disquisitiones sequentes incomparabiter contrahere

licuisset; . . .

. . . but nothing prevents us from decomposing ξ, nevertheless, into fac-

tors of two, three or more dimensions [degrees], whereupon, in some

sense, imaginary roots could be attributed to them. Indeed, we could

have shortened incomparably all our following investigations, had we

wanted to introduce such imaginary quantities by taking the same lib-

erty some more recent mathematicians have taken; . . .

Normat 2/2009 David A. Cox 53

Over the complex numbers, Gauss was the ﬁrst to prove the existence of roots of

polynomials. He was critical of those who simply assumed that roots exist, so he

clearly wasn’t going to assume that congruences of higher degree have solutions.

We refer the reader to [11] for a fuller account of Gauss’s work on ﬁnite ﬁelds.

Unfortunately, none of this was available until after Gauss’s death in 1855. In

particular, Schönemann was unaware of these developments when he rediscovered

many of Gauss’s results in the 1840s.

Abel

Gauss’s cryptic comments about the lemniscate in Disquisitiones had a profound

inﬂuence on Abel. He developed the theory of elliptic functions (as did Jacobi),

based on the equation

= (1 − c

)(1 + e

), (1)

and his elliptic functions were inverse functions to the elliptic integral

(1 − c

)(1 + e

)

When e = c = 1, we get the integral associated with the lemniscate.

The division problem for elliptic integrals goes back to Fagnano and Euler. Later

in the article we will quote a letter from Eisenstein to Gauss, where he expressed

the m-division problem in the lemniscatic case as the “algebraic integral of the

equation

dy/

1 − y

= m

dx/

1 − x

.” (2)

In modern language, we are talking about division points on elliptic curves, and the

“algebraic integral” produces a polynomial P

of degree m

whose solutions give

(roughly speaking) the m-division points on the associated elliptic curve deﬁned

by (1). We will say more about the polynomial P

and the equation (2) when we

discuss Eisenstein.

For Abel and his contemporaries, a central question was whether polynomial

equations such as P

(x) = 0 were “solvable algebraically”, which these days means

solvable by radicals. Abel was uniquely qualiﬁed to pose this question, since just

four years earlier he had proved that the general quintic was not solvable by radicals.

In his great paper Recherches sur les functions elliptiques [1, pp. 263–388],

printed in volumes 2 and 3 of Crelle’s journal

in 1827 and 1828, Abel consid-

ers the equation P

2n+1

= 0 coming from (2n + 1)-division points on the elliptic

curve (1). Here is what he has to say about this equation:

Donc en dernier lieu la résolution d’équation P

2n+1

= 0 est reduite

á celle d’une seule équation du degré 2n + 2; mais en général cette

équation ne paraît pas être résoluble algébriquement. Néamoins on peut

las résoudre complètement dans plusiers cas particuliers, par exemple,

lorsque e = c, e = c

√

3, e = c(2±

√

3) etc. Dans le course de ce mémoire

The Journal für die reine und angewandte Mathematik, founded by August Leopold Crelle

in 1826.

54 David A. Cox Normat 2/2009

je m’occuperai de ces cas, dont le premier surtout est remarquable, tant

pour la simplicité de la solution, que par sa belle application dans la

géométrie.

En eﬀet entre autres théorèmes je suis parvenu à celui-ci:

“On peut diviser la circonférence entière de la lemniscate en m parties

égales par la regle et le compas seuls, si m est de la forme 2

ou 2

+ 1,

ce dernier nombre étant en même temps premier; ou bien si m est un

product de plusiers nombres de ces deux formes.”

Ce théorème est, comme on le voit, précisément le même que celui de

M. Gauss, relativement au cercle.

Thus ﬁnally the solution of the equation P

2n+1

= 0 is reduced to

a single equation of degree 2n + 2; but in general this equation does

not appear to be solvable algebraically. Nevertheless one can solve it

completely in many particular cases, for example, when e = c, e = c

√

e = c(2 ±

√

3) etc. In the course of this memoir I will concern myself

with these cases, of which the ﬁrst is especially remarkable, both for

the simplicity of its solution, as well as by its beautiful application to

geometry.

Indeed among other theorems I arrived at this one:

“One can divide the entire circumference of the lemniscate into m parts

by ruler and compass only, if m is of the form 2

or 2

+ 1, the last

number being at the same time prime, or if m is a product of several

numbers of these two forms.”

This theorem is, as one sees, precisely the same as that of M. Gauss,

relative to the circle.

The reduction to an equation of degree 2n + 2 was done by classical methods of

Lagrange. Besides the mind-blowing result about the lemniscate (e = c), other

aspects of this quote deserve comment:

• The cases e = c, e = c

√

3, e = c(2 ±

√

3) etc. that Abel can solve by radi-

cals correspond to elliptic curves with complex multiplication (see [5] for an

introduction). Abel was the ﬁrst to identify this important class of elliptic

curves.

• From a modern standpoint, division points of elliptic curves with complex

multiplication generate Abelian extensions and hence have Abelian Galois

groups. Since Abelian groups are solvable, Galois theory implies that the

extensions are solvable by radicals.

• When the curve doesn’t have complex multiplication, Abel was more cautious:

they do “not appear to be solvable algebraically.” By deep work of Serre on

Galois representations of elliptic curves [27], we now know that with at most

ﬁnitely many exceptions, these equations aren’t solvable by radicals.

Normat 2/2009 David A. Cox 55

Again we are in the presence of remarkably rich mathematics.

Abel thought deeply about why his equations P

2n+1

= 0 were solvable by radicals

when the curve has complex multiplication. He realized that the underlying reason

was the structure of the roots and how they relate to each other. His general result

appears in his Mémoire sur une classe particulière d’équations résolubles algébri-

quement [1, pp. 478–507], which was published in Crelle’s journal in 1829. The

article begins:

Quoique la résolution algébrique des équations ne soit possible en

général, il y a néamoins des équations particulières des tous les degrés

qui admettant une telle résolution. Telles sont par example les équations

de la forme x

− 1 = 0. La résolution de ces équations est fondée sur

certaines relations qui existent entre les racines.

Although the algebraic solution of equations is not possible in general,

there are nevertheless particular equations of all degrees which admit

such a solution. Examples are the equations of the form x

− 1 = 0.

The solution of these equations is based on certain relations that exist

among the roots.

The ﬁrst sentence refers to Abel’s result on the unsolvability of the general quintic

and the solution of x

− 1 = 0 described by Gauss in Disquisitiones. To give the

reader a sense of what he means by “relations that exist among the roots,” Abel

takes a prime n and considers the cyclotomic equation x

n−1

+ ··· + x + 1 = 0.

Deﬁne the polynomial θ(x) = x

, where α is a primitive root modulo n. Then the

roots are given by

x, θ(x) = x

, θ

(x) = x

, θ

(x) = x

, . . . , θ

n−2

(x) = x

n−2

, where θ

n−1

(x) = x.

Abel goes on to say that the same property appears in a certain class of equations

that he found in the theory of elliptic functions. He then states the main theorem

of the paper:

En général j’ai démontré le thèoréme suivant:

,,Si les racines d’une équation d’un degré quelquonque sont liées entre

elles de telle sorte, que toutes ces racines puissent être exprimées ra-

tionnellement au moyen de l’une d’elles, que nous désignerons par x; si

de plus, en désignant par θx, θ

x deux autres racines quelquonques, on

θθ

x = θ

θx,

l’équation dont il s’agit sera toujours résoluble algébriquement. . . . ”

In general I have proved the following theorem:

,,If the roots of an equation of arbitrary degree are related among them-

selves in such a way, that all of the roots can be rationally expressed

56 David A. Cox Normat 2/2009

in terms of one of them, which we designate by x; if in addition, desig-

nating by θx, θ

x two other arbitrary roots, one has

θθ

x = θ

θx,

the equation in question is always solvable algebraicially. . . . ”

Abel’s “classe particulière” consist of all polynomials that satisfy the hypothesis of

his theorem. To see what this says in modern terms, let K ⊆ L be a Galois extension

with primitive element α. For each element σ

of the Galois group Gal(L/K), there

is a polynomial θ

(x) ∈ K[x] such that σ

(α) = θ

(α). Then one easily computes

that

(α) = θ

(θ

(α)).

The switch of indices is correct—you should check why. Since σ

is determined by

its value on α,

= σ

⇐⇒ θ

(θ

(α)) = θ

(θ

(α)).

Since the θ

(α) are the roots of the minimal polynomial f(x) of α over K, we see

that f(x) is in the “classe particulière” if and only if Gal(L/K) is commutative. As

noted earlier, this means that the Galois group is solvable, so that f(x) is solvable

by radicals by Galois theory.

Besides proving his general theorem, Abel intended to give two applications:

Après avoir exposé cette théorie en général, je l’appliquerai aux fonc-

tions circulares et elliptiques.

After having developed this theory in general, I will apply it to circular

and elliptic functions.

The version published in Crelle’s journal has a section on circular functions, but

ends with the following footnote by Crelle:

*) L’auteur de ce mémoire donnera dans une autre occasion des

applications aux fonctions elliptiques.

*) The author of this memoir will give applications to elliptic func-

tions on another occasion.

Alas, Abel died shortly after this article appeared.

After Abel

Abel’s “classe particulière” had an important inﬂuence on Kronecker, Jordan, and

Weber. Speciﬁcally:

• In 1853, Kronecker [18, Vol. IV, p. 11] deﬁned f(x) = 0 to be “Abelian”

provided it has roots x, θ(x), . . . , θ

n−1

(x), x = θ

(x). Here, as for Abel, θ is a

rational function. This special case of Abel’s “classe particulière” corresponds

to polynomials with cyclic Galois groups.

Normat 2/2009 David A. Cox 57

• In 1870, Jordan [17, p. 287] deﬁned f(x) = 0 to be “Abelian” in terms of its

Galois group:

Nous appellerons donc équations abéliennes toutes celles dont le

groupe ne contient que les substitutions échangeables entre elles.

We thus call Abelian equations all of those whose group only con-

tains substitutions that are exchangeable among each other.

Here, “exchangeable” is Jordan’s way of saying “commutative.” He then proves

[17, p. 288] that for irreducible equations, his deﬁnition is equivalent to Abel’s

“classe particulière.”

• The ﬁrst two volumes of Weber’s monumental Lehrbuch der Algebra were

published in 1894 and 1896. He gives the name “Abelian” to Abel’s “classe

particulière” [29, Vol. I, p. 576] and later deﬁnes a commutative group to be

“Abelian” [29, Vol. II, p. 6]. As far as I know, this is the ﬁrst appearence of

the term “Abelian group” in the modern sense.

The deﬁnition of “Abelian group” given in introductory algebra courses seems so

simple. But in the background is a rich history involving Gauss, Abel, the leminis-

cate, elliptic functions, complex multiplication, and solvability by radicals.

Galois

One of the few papers published during Galois’s lifetime was Sur la théorie des

nombres, appearing in 1830 in the Bulletin des sciences mathématiques de Ferussac

[12, pp. 113–127]. This paper develops the theory of ﬁnite ﬁelds. Galois begins with

a congruence F (x) ≡ 0 mod p, or as he writes it, Fx = 0, where F (x) is irreducible

modulo p. Then he considers the roots:

. . . Il faut donc regarder les racines de cette congruence comme des

espèces de symboles imaginaires . . .

. . . One must regard the roots of this congruence as a kind of imaginary

symbol . . .

It is clear that Gauss would not approve. Galois used the symbol i to denote a root

of F (x) ≡ 0 mod p, and he showed that the numbers

a + a

i + a

+ ··· + a

ν−1

where ν = deg(F ) and a, a

, . . . , a

ν−1

are integers modulo p, form a ﬁnite ﬁeld

with p

elements. Galois went on to develop a complete theory of ﬁnite ﬁelds. The

reason he needed ﬁnite ﬁelds is connected with his deep work on the structure of

solvable primitive permutation groups (see the Historical Notes to [4, §14.3]).

We will not say more about Galois and ﬁnite ﬁelds, because Schönemann was

not aware of Galois’s 1830 paper when he began his own study of congruences and

ﬁnite ﬁelds in the early 1840s.

In 1870, Jordan used the term “groupe abélien” to refer to a group closely related to a

symplectic group over a ﬁnite ﬁeld [17, Livre II, §VIII].

58 David A. Cox Normat 2/2009

Schönemann

Unlike the other people mentioned so far, Theodor Schönemann is not a famil-

iar name. He has no biography at the MacTutor History of Mathematics archive

[21]. According to the Allgemeine Deustsche Biographie [2, Vol. 32, pp. 293–294],

Schönemann lived from 1812 to 1868 and was educated in Königsberg and Berlin

under the guidance of Jacobi and Steiner. He got his doctorate in 1842 and be-

came Oberlehrer and eventually Professor at a gymnasium in Brandenburg an der

Havel. Lemmermeyer’s book [19] includes several references to Schönemann’s work

in number theory, and some of his results are mentioned in Dickson’s classic History

of the Theory of Numbers [7], especially in the chapter on higher congruences in

Volume I.

For us, Schönemann’s most important work is a long paper printed in two parts in

Crelle’s journal in 1845 and 1846. The ﬁrst part [24], consisting of §1–§50, appeared

as Grundzüge einer allgemeinen Theorie der höhern Congruenzen, deren Modul

eine reelle Primzahl ist (Foundations of a general theory of higher congruences,

whose modulus is a real prime number). In the preface, Schönemann refers to Gauss:

Der berühmte Verfasser der Disquisitiones Arithmeticae hatte für

den achten abschnitt seines Werkes eine allgemeine Theorie der höh-

ern Congruenzen bestimmt. Da indessen dieser achte Abschnitt nicht

erscheinen, und auch, so viel ich weiss, über diesen Gegenstand sonst

nichts von dem Herrn Verfasser bekannt gemacht oder nur bestimmt

angedeutet worden ist . . .

The famous author of Disquisitiones Arithmeticae had intended a

general theory of higher congruences for Section Eight of his work.

Since, however, this Section Eight did not appear, and also, as far as I

know, the author did not publish anything on this subject, nor indicate

anything precisely . . .

Schönemann suspects that he may have been scooped by Gauss, but is not worried:

. . . würde mich über die Einbusse der ersten Enteckung das Bewusstsein

schadlos halten, auf selbständigem Wege mit dem Streben eines solchen

Geistes zusammengetroﬀen zu sein.

. . . the loss of ﬁrst discovery would be compensated by my knowing of

having met in my own and independent way such a spirit.

Indeed, Schönemann had been scooped by both Gauss and Galois. Hence we should

change “a spirit” to “spirits” in the quote, in which case the sentiment is even more

apt.

Similar to what Gauss did, Schönemann gave a careful treatement of polynomi-

als modulo p, including unique factorization. But then, in §14, he did something

diﬀerent. Let f (x) ∈ Z[x] be monic of degree n and irreducible modulo p, and let

α ∈ C be a root of f (x) (Gauss would approve of this root). Then, given polynomi-

als ϕ, ψ ∈ Z[x], Schönemann deﬁned ϕ(α) and ψ(α) to be congruent modulo (p, α)

if ϕ(α) = ψ(α) + pR(α) for some R ∈ Z[x]. He also proved:

Normat 2/2009 David A. Cox 59

• The “allgemeine Form eines kleines Restes” (“general form of a smallest re-

mainder”) is a

n−1

+ a

n−2

+ ··· + a

n−1

, where a

∈ {0, . . . , p − 1}. This

gives the ﬁnite ﬁeld F

• The elements of F

are the roots of x

− x ≡ 0 mod (p, α).

• f(x) ≡ (x − α)(x − α

) ···(x − α

n−1

) mod (p, α). Thus F

is the splitting

ﬁeld of f(x) mod (p, α). Also, the Galois group (generated by Frobenius) is

implicit in this factorization of f.

The ﬁrst part of Schönemann’s paper culminates in §50 with a lovely proof of the

irreducibility of Φ

(x) = x

p−1

+ ··· + x + 1. We will give the proof in modern

notation. Pick a prime ` 6= p and consider the prime factorization

(x) ≡ f

(x) ···f

(x) mod `.

where the f

are irreducible modulo `. Standard properties of ﬁnite ﬁelds imply

that

deg(f

) = the minimum n such that F

∗

has an element of order p

= the minimum n such that `

≡ 1 mod p

= the order of the congruence class of ` in (Z/pZ)

∗

(3)

We leave this as a fun exercise for the reader. By Dirichlet’s theorem on primes

in arithmetic progressions (proved just a few years before Schönemann’s paper),

every congruence class modulo p contains a prime. In particular, the congruence

class of a primitive root contains a prime `. A primitive root modulo p gives a

congruence class of order p − 1 in (Z/pZ)

∗

, so that n = p −1 in (3) for this choice

of `. This implies that Φ

(x) is irreducible modulo ` and hence irreducible over Z.

Then Φ

(x) is irreducible over Q by Gauss’s Lemma.

This proof is simpler than Gauss’s, though it does require knowledge of ﬁ-

nite ﬁelds. The use of the auxiliary prime ` is especially elegant. When I studied

Grothendieck-style algebraic geometry as a graduate student in the 1970s, I was

always happy when a proof picked a prime diﬀerent from the residue characteristic.

This seemed so modern and cutting-edge. Little did I realize that Schönemann had

used the same idea 120 years earlier.

The second part of Schönemann’s paper [25], titled Von denjenigen Moduln,

welche Potenzen von Primzahlen sind (On those moduli, which are powers of prime

numbers), consists of §51–§66. In this paper, Schönemann considered the factoriza-

tion of polynomials modulo p

, and in particular, how the factorization changes

as m varies. One of his major results, in §59, is what we now call Hensel’s Lemma:

Lehrsatz. Ist irgend ein einfacher Ausdruck von x nach dem Modul p

in zwei einfache Factoren zerlegbar, die nach demselben Mdoul keinen

gemeinsaftlichen Divisor haben: so ist dieser Ausdruck auch nach dem

Modul p

, aber nur auf einer Weise, in zwei Factoren zerlegbar,

welche jenen beiden ersten nach dem modul p congruent sind.

60 David A. Cox Normat 2/2009

Lemma. If any monic polynomial of x can be factored modulo p into

two monic factors, which for this modulus have no common divisor: then

this polynomial can be factored modulo p

, in a unique manner, into

two factors, which are congruent to those ﬁrst two factors modulo p.

(Here, “einfacher Ausdruck von x” means a monic polynomial of x.) As a conse-

quence, when an irreducible polynomial modulo p

is reduced modulo p, the result

must be a power of an irreducible polynomial modulo p. In §61, Schönemann asks

about the converse:

Aufgabe. Zu untersuchen, ob die Potenz eines nach dem Modul p

irreductibeln Ausdrucks, nach dem Modul p

irreducibel sei, oder nicht.

Problem. To investigate, whether the power of irreducible polynomial

modulo p is or is not irreducible modulo p

An especially simple example is (x−a)

, and for a polynomial congruent to (x−a)

modulo p, the ﬁrst place to check for irreducibility is modulo p

. Here is Schöne-

mann’s answer:

. . . man darf daher den Satz aussprechen: dass (x −a)

+ pFx nach

dem Modul p

irreductibel sein wurde, wenn Fx nach dem

Modul p nicht den Factor x −a in sich schliesst. . . .

. . . hence one may state the theorem: that (x −a)

+ pFx is irre-

ducible modulo p

, when the factor x −a is not contained in

Fx modulo p. . . .

As stated, this is not quite correct—one needs to assume that deg(F ) ≤ n.

Since

x − a divides F(x) modulo p if and only if F(a) ≡ 0 mod p, we can state Schöne-

mann’s result as follows.

Schönemann’s Criterion. Let f(x) ∈ Z[x] have degree n > 0 and

assume that there is a prime p and an integer a such that

f(x) = (x −a)

+ pF (x),

If F(a) 6≡ 0 mod p, then f(x) is irreducible modulo p

The proof is not diﬃcult (assume (x−a)

+pF (x) factors modulo p

, reduce modulo

p and use unique factorization in F

[x]) and is left to the reader.

The pleasant surprise is that this result implies the Eisenstein criterion. To see

why, suppose that f(x) = a

+ a

n−1

+ ··· + a

satisﬁes the hypothesis of

The uniqueness assertion enables us to take the limit as m → ∞, giving a factorization over

the p-adic integers that reduces to the given factorization modulo p. This version of Hensel’s

Lemma is stated in [16, Thm. 3.4.6], and the discussion on [16, p. 72] explains how this relates to

the more common version of Hensel’s Lemma, which asserts that for f(x) ∈ Z

[x], a solution of

f(x) ≡ 0 mod p of multiplicity one lifts to a solution of f(x) = 0 in Z

For example, let F (x) = x

− p

x + 1. Then x

+ pF (x) = (px + 1)(x

− p

x + p), yet x does

not divide F (x) modulo p.

Normat 2/2009 David A. Cox 61

the Eisentstein criterion. Multiplying by a suitable integer, we may assume a

≡

1 mod p. This allows us to write f(x) = x

+pF (x). Note also that F (0) 6≡ 0 mod p

since p

does not divide a

. Then f(x) is irreducible modulo p

by Schönemann’s

criterion. This implies irreducibility over Z and hence (via Gauss’s Lemma) over

As you might expect, Schönemann immediately applies his irreducibility crite-

rion to a familiar polynomial:

Wenden wir das erhaltene Resultat auf der Ausdruck

− 1

x − 1

an, wo

n eine Primzahl bedeutet. Es ist für diesen Fall x

−1 ≡ (x−1)

(mod.

n), und man erhält also

− 1

x − 1

= x

n−1

+ x

n−2

+ ···· + x + 1 = (x − 1)

n−1

+ nF x.

Für x = 1 erhält man n = nF (1) und daher F (1) = 1, und nicht ≡ 0

(mod. n). Hieraus folgt, dass

− 1

x − 1

nach dem Modul n

stets

irreductibel ist, wenn n eine Primzahl bedeutet; mithin muss

dieser Ausdruck gewiss in algebraischer Beziehung irreductibel

sein.

Die Leichtigkeit des Beweises dieses Satzes is auﬀallend, da derselbe in

den ,,Disquisitiones” mit einem viel grössern Aufwande von Scharfsinn,

und dennoch viel umständlicher geführt is. (Vergl. §. 50. Zus. 2.)

Let us apply the result just obtained to the polynomial

− 1

x − 1

, where

n denotes a prime number. In this case x

−1 ≡ (x −1)

(mod. n), and

one thus obtains

− 1

x − 1

= x

n−1

+ x

n−2

+ ···· + x + 1 = (x − 1)

n−1

+ nF x.

For x = 1 one obtains n = nF(1) and thus F (1) = 1, and not ≡ 0

(mod. n). From this, it follows that

− 1

x − 1

is always irreducible

modulo n

, if n is a prime number; hence, this expression is

certainly irreducible in the algebraic sense.

The ease of proof of this theorem is striking, because the proof in

,,Disquisitiones” requires much greater cleverness, and is much more

elaborate. (See §. 50. Rem. 2.)

This proves the irreducibility of x

n−1

+ ··· + x + 1 without the change of variable

x ↔ x + 1 needed when one uses the Eisenstein criterion. Schönemann is clearly

pleased that his proof is so much simpler than Gauss’s. (The parenthetical comment

at the end of the quote refers to Schönemann’s earlier proof of irreducibility given

in §50 of the ﬁrst part of his article.)

62 David A. Cox Normat 2/2009

Schönemann’s criterion is lovely but is unknown to most mathematicians. So

how did I learn about it? My book on Galois theory [4] gives Eisenstein’s proof

of Abel’s theorem on the lemniscate. In trying to understand Eisenstein, I looked

at Lemmermeyer’s wonderful book Reciprocity Laws, where I found a reference

to Schönemann. When I tried to read Schönemann’s paper, I couldn’t ﬁnd the

Eisenstein criterion, in part because the paper is long and my German isn’t very

good, and in part because I was looking for Eisenstein’s version, not Schönemann’s.

I looked back at Lemmermeyer’s book and noticed that Lemmermeyer thanked

Michael Filaseta for the Schönemann reference. I wrote to Filaseta, who replied

that Schönemann proved a criterion for a polynomial to be irreducible modulo p

This quickly led me to §61 of the article, which is where Schönemann states his

result.

Back to Gauss

Besides discovering the Eisenstein criterion before Eisenstein, Schönemann also

discovered Hensel’s Lemma before Hensel. Unfortunately, Schönemann and Hensel

were both scooped by Gauss. In his draft of the unpublished eighth section of

Disquisitiones (p. 627 of the German version of [13] or p. 238 of [15, Vol. II]), Gauss

takes a polynomial X with integer coeﬃcients and studies its behavior modulo p

and p

Problema. Si functio X secundum modulum p in factores inter se

primos ξ, ξ

, ξ

etc. sit resoluta, X secundum modulum pp in similes

factores Ξ, Ξ

, Ξ

etc. resolvere ita, ut sit

ξ ≡ Ξ, ξ

≡ Ξ

, ξ

≡ Ξ

, etc. (mod.p)

Problem. If the polynomial X decomposes modulo p into mutually

prime factors ξ, ξ

, ξ

etc., then similarly X decomposes modulo p

into factors Ξ, Ξ

, Ξ

etc. such that

ξ ≡ Ξ, ξ

≡ Ξ

, ξ

≡ Ξ

, etc. (mod.p)

Gauss proves this and then explains how the same principle applies modulo p

for any k. His “Problema” is weaker than Schönemann’s “Lemma” because it

doesn’t say that the lifted factorization is unique. So what Gauss really proved was

a “proto-Hensel’s Lemma.” Nevertheless, Gauss was suﬃciently pleased with this

result that he recorded it in his famous mathematicial diary [14]. Here is entry 79,

dated September 9, 1797:

Principia detexi, ad quae congruentiarum secundum modulos multi-

plices resolutio ad congruentias secundum modulum linearem reduci-

tur.

Beginning to uncover principles, by which the resolution of congruences

according to multiple moduli is reduced to congruences according to

linear moduli.

Normat 2/2009 David A. Cox 63

Here, “resolution of congruences according to multiple moduli” means factoring

polynomials modulo p

, and similarly “congruences according to linear moduli”

means working modulo p. This reading of Gauss’s entry is carefully justiﬁed in

[11].

Besides this elementary version of Hensel’s Lemma, Gauss also considered the

case when the factors modulo p are not distinct. For example, the congruence

X ≡ X

(x−a)

mod p appears near the end of Gauss’s draft of the eighth section.

Had he pursued this, it is quite possible that he would have followed the same path

as Schönemann and discovered the Eisenstein criterion. But instead, the draft ends

abruptly in the middle of a congruence: the last thing Gauss wrote was

0 ≡

As with many other projects, Gauss never returned to ﬁnish Disquisitiones gen-

erales de congruentiis. It came to light only after being published in 1863 in the

second volume of his collected works, and today is still overshadowed by its more

famous sibling, Disquisitiones Arithmeticae.

After Schönemann

Although Schönemann was scooped on ﬁnite ﬁelds by Gauss and Galois, he went

beyond both of them in one signiﬁcant way: he gave a rigorous description of

the elements of a ﬁnite ﬁeld. Gauss would have been very critical of the roots

of congruences so blithely assumed by Galois. Schönemann, by starting with a

complex root α ∈ C of a monic polynomial f (x) that is irreducible modulo p,

constructed the ﬁeld whose modern description is the quotient ring Z[α]/hpi, where

hpi is the ideal of Z[α] generated by p.

Schönemann’s construction, while rigorous, is not purely algebraic, since it de-

pends on the root α ∈ C of f(x). This uses the Fundamental Theorem of Algebra,

which in spite of its name is a theorem in analysis since it ultimately depends on

the completeness of the real numbers. Of course, these days, we would express Z[α]

via the isomorphism

Z[X]/hf(X)i ' Z[α]

induced by X 7→ α, so that our ﬁnite ﬁeld is

Z[X]/hp, f(X)i ' Z[α]/hpi.

This algebraic version of ﬁnite ﬁelds was made explicit by Dedekind in his 1857

paper Abriß einer Theorie der höheren Kongruenzen in bezug auf einen reellen

Primzahl-Modulus (Outline of a theory of higher congruences for a real prime mod-

ulus) [6]. Dedekind begins the paper by noting that the subject was initiated by

Gauss and had been studied by Galois and Schönemann. Dedekind was at the time

unaware of the full power of what Gauss had done, though later he became the

editior in charge of publishing Disquisitiones generales de congruentiis in Volume

II of Gauss’s collected works in 1863.

Dedekind’s construction is essentially what we did above with the quotient ring

Z[X]/hp, f(X)i, f(X) irreducible modulo p, though Dedekind was writing before

the concept of quotient ring was fully established. Nevertheless, he shows that this

64 David A. Cox Normat 2/2009

is a ﬁnite ﬁeld with p

elements, n = deg(f). For much of the 19th century, “ﬁnite

ﬁeld” meant this object. It has the advantage of being easy to compute with (even

today, computers represent ﬁnite ﬁelds this way), but mathematically, it depends

on the choice of f(X) and hence is intrinsically non-canonical.

One of the ﬁrst fully abstract deﬁnitions of ﬁnite ﬁeld was given by E. H. Moore,

whose paper [20] appeared in the proceedings of the 1893 international congress of

mathematicians. Here is his deﬁnition:

Suppose that we have a system of s distinct symbols or marks

∗

, . . . , µ

(s being some ﬁnite positive integer), and suppose that these

marks may be combined by the four fundamental operations of algebra—

addition, subtraction, multiplication, and division—these operations

being subject to the ordinary abstract operational identities of algebra

+ µ

= µ

+ µ

; µ

= µ

; (µ

+ µ

)µ

= µ

+ µ

; etc.

and that when the marks are so combined the results of these operations

are in every case uniquely determined and belong to the system of

marks. Such a system we shall call a ﬁeld of order s, using the notation

F [s].

We are led at once to seek To determine all such ﬁelds of order s, F[s].

The words “system” and “marks” indicate that Moore was writing before the lan-

guage of set theory was standardized. Moore went on to show that his deﬁnition

was equivalent to the Dedekind-style representation of a ﬁnite ﬁeld. So in 1893 we

ﬁnally have a modern theory of ﬁnite ﬁelds.

The word “marks” in Moore’s quote has an the asterisk that leads to the follow-

ing footnote:

∗

It is necessary that all quantitative ideas should be excluded from the

concept marks. Note that the signs >, < do not occur in the theory.

Moore was writing for a mathematically sophisticated audience, but he didn’t as-

sume that they had the apparatus of set theory in their heads—his footnote was

intended to help them understand the abstract nature of what he was saying. This

is something we should keep in mind when we teach abstract algebra to undergrad-

uates.

Eisenstein

We ﬁnally get to Eisenstein, whose work on Abel’s theorem on the lemniscate

culminated in a long two-part paper in Crelle’s journal in 1850 [10, pp. 536–619].

Eisenstein used Abel’s notation ϕ for the lemniscatic function, so that

r = ϕ(s) ⇐⇒ s =

√

1 − r

. (4)

(We follow the 19th century practice of using the same letter for the variable and

limit of integration.) In this equation, 0 ≤ r ≤ 1 corresponds to 0 ≤ s ≤ $ =

Normat 2/2009 David A. Cox 65

1−1

Figure 1: Arc length on the lemniscate

√

1−r

. Then deﬁne ϕ for s ≥ 0 by considering the point on the lemniscate whose

cumulative arc length is s when we start from the origin and follow the branch of

the lemniscate in the ﬁrst quadrant. See Figure 1. An arc length calculation shows

that s and the radius r are related by the equation

r = ϕ(s)

(see [4, §15.2]). In particular, $ is one-fourth of the total arc length of the lem-

niscate, so that ϕ($) = 1 and ϕ(2$) = 0. Hence, for any positive integer m,

r = ϕ(k · 2$/m), k = 1, . . . , m, gives the radii of the points that divide the right

half of the lemniscate into m equal pieces.

The change of variables r = iu in (4) led Abel to deﬁne ϕ(is) = iϕ(s), and

then Euler’s addition law makes ϕ(z) = ϕ(s + it) into a function of a complex

variable z ∈ C.

A key observation is that for any Gaussian integer m ∈ Z[i],

ϕ(mz) is a rational function of ϕ(z) and its derivative ϕ

(z). This is what complex

multiplication means for the lemniscatic function ϕ.

When m = a + ib is odd Gaussian integer, meaning that a + b is odd, ϕ(mz) is

a rational function of ϕ(z) alone. Here, one can ﬁnd polynomials U(x), V (x) with

coeﬃcients in Z[i] such that

y = ϕ(mz) is related to x = ϕ(z)

via

y =

U(x)

V (x)

x + A

+ ··· + A

(N(m)−1)/4

N(m)

1 + B

+ ··· + B

(N(m)−1)/4

N(m)−1

(5)

where N (m) = a

+ b

. See [4, Thm. 15.4.4] for a proof.

When m is an ordinary odd integer, we know that r = ϕ(k · 2$/m) gives m-

division points on the lemniscate. Setting

y = ϕ(m · (k · 2$/m)) = ϕ(k · 2$) = 0 and x = ϕ(k · 2$/m) = r

into (5), we see that

0 =

U(r)

V (r)

, hence U (r) = 0.

Gauss followed the same path in 1797, though he never published his ﬁndings. See [3] for

more details.

66 David A. Cox Normat 2/2009

This proves that the division radii r are roots of the polynomial equation U(x) = 0.

When m = 2n + 1, this is precisely the equation P

2n+1

(x) = 0 considered by Abel.

To prove Abel’s theorem, one can reduce to the case when m = a + ib is an odd

Gaussian prime. Since U(x) has x as a factor. Eisenstein wrote U(x) = xW (x),

and the strategy of his proof was to show that W (x) is irreducible. Once we know

this, Abel’s theorem follows—see [4, §15.5].

But how do you prove that a polynomial such as W (x) is irreducible? This

is not easy. A key step for Eisenstein was when he noticed something about the

coeﬃcients of W(x). He shared his thoughts with Gauss in a letter dated 18 August

1847 [10, p. 845]. Before quoting the letter, we need to observe that in terms of

integrals, the equations y = ϕ(mz) and x = ϕ(z) imply that

1 − y

= m

√

1 − x

In 19th century parlance, the relation between y and x given by (5) is an algebraic

integral of this equality of integrals. Now the quote:

Wenn m = a + bi eine ungerade complexe Zahl, p deren Norm und

y =

x + A

+ ···· +A

(p−1)/4

1 + B

+ ···· +B

(p−1)/4

p−1

das algebraische Integral der

Gleichung

dy/

1 − y

= m

dx/

1 − x

ist, so hatte ich früher gezeigt, daß für eine zweigliedrige complexe

Primzahl m die Coeﬃcienten des Zählers bis auf den letzten, welcher

eine complexe Einheit ist, und die Coeﬃcienten des Nenners bis auf

den Ersten, welcher = 1, alle durch m theilbar sind. Ich vermuthete,

daß der Satz auch richtig sei, wenn m eine eingliedrige Primzahl (≡ 3

(mod 4) abgesehen vom Zeichen oder von einer complexen Einheit als

Factor) ist;

When m = a + bi is an odd complex integer of norm p and y =

x + A

+ ···· +A

(p−1)/4

1 + B

+ ···· +B

(p−1)/4

p−1

is the algebraic integral of the equation

dy/

1 − y

= m

dx/

1 − x

so I had earlier shown that for a two-term complex prime number m

the coeﬃcients of the numerator up to the last, which is a complex unit,

and the coeﬃcients of the denominator except the ﬁrst, which = 1, are

all divisible by m. I conjectured that this proposition is also correct

when m is a one-term prime number (≡ 3 (mod 4) apart from sign or

a complex unit as factor);

For a complete proof of Abel’s theorem on the lemniscate, the reader should consult [4], [22]

or [23]. The last reference gives a modern proof via class ﬁeld theory.

Normat 2/2009 David A. Cox 67

(Note the use of four dots instead of three.) In the ﬁrst part of the quote, Eisenstein

sets up the situation, and after the displayed equation, describes the structure of

the coeﬃcients of the numerator and denominator. Odd Gaussian primes come in

two ﬂavors:

• Two-term primes of the form m = a + ib, where p = a

+ b

is prime and

p ≡ 1 mod 4.

• One-term primes of the form m = εq, where ε is a unit in Z[i] and q ≡

3 mod 4.

Now consider the polynomial

W (x) =

U(x) = A

+ A

+ ··· + A

(p−1)/4

p−1

For a two-term prime m, Eisenstein says that he earlier had shown that the last

coeﬃcient A

(p−1)/4

is a complex unit and the other coeﬃcients A

, . . . , A

(p−1)/4−1

are divisible by m. He conjectures that the same is true for one-term primes.

This smells like the Eisenstein criterion, especially since Eisenstein notes in the

letter that the constant term A

is m, which is not divisible by m

. The diﬀerence

is that m and the coeﬃcients of W are Gaussian integers. A bit later in the letter,

Eisenstein considers what happens if W is not irreducible over Q(i) [10, pp. 848–

849]:

. . . wenn es möglich ist W das Produkt aus zwei rationalen ganzen Funk-

tionen von x mit ganzen complexen Coeﬃcienten, und deren Grade

< p − 1 sind. Es sei W = PQ; da das constante Glied von W , = m

ist, so kann, wenn m eine complexe Primzahl ist, das Constante Glied

in einer der beiden ganzen Funktionen P, Q nur = 1, in der anderen

= m sein; denn die Coeﬃcienten in P und Q müssen, wenn sie rational

sind, nothwendig ganz sein, wie man durch dieselben Betrachtungen

nachweisen kann, welche Ew. Hochwohlgeboren schon in der reellen

Zahlentheorie (Disq. Sectio prima) angestellt haben.

. . . if it is possible that W is the product of two polynomials of x with

Gaussian integer coeﬃcients, and their degrees are < p − 1. Let W =

P Q; since the constant term of W is = m, so if m is a complex prime,

the constant term in one of the two polynomials P, Q is = 1 and the

other = m; then the coeﬃcients of P and Q if rational, must necessarily

be integral, as one can show by the same considerations which your

Eminence

used in the real number theory (Disq. Section I).

The literal translation of “Ew. Hochwohlgeboren” is “your High Well Born,” which sounds silly

in English. So I used “your Eminence” instead. The word “Hochwohlgeboren” originally applied to

lesser German nobility and gentry. This ﬂowery language is reﬂected in the letter’s saluation, “Sr.

Hochwohlgeboren, dem Geheimrath pp. Prof. Dr. Gauss”, which translates “To his Eminence,

the Distinguished, and so on, Professor Doctor Gauss.” The word “Geheimrath,” now spelled

“Geheimrat,” originated as the German equivalent of a “Privy councillor” in a governmental

context and was an honoriﬁc for distinguished professors in German universities in the 19th

century.

68 David A. Cox Normat 2/2009

Here, “real number theory” means over Z rather than Z[i], and the reference to

Disquisitiones is the ﬁrst Gauss quote of this article. Thus Eisenstein is telling

Gauss that Gauss’s Lemma applies to the Gaussian integers. Mind-blowing. Then

Eisenstein proceeds to prove that W is irreducible using one of the standard proofs

of the Eisenstein criterion.

In other words, Eisenstein’s ﬁrst proof of his criterion

• was over the Gaussian integers;

• applied to a polynomial associated with the division problem on the lemnis-

cate; and

• appeared in a letter to Gauss.

When Eisenstein wrote up his results for publication, he realized that his criterion

was much more general. The ﬁrst part of his long paper had the title Über die Irre-

ducibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung

der ganzen Lemniscate abhängt (On the irreducibility and some other properties of

equations that depend on the division of the lemniscate) [10, pp. 536–555]. This

paper contains Eisenstein’s version of the Eisenstein criterion:

,,Wenn in einer ganzen Funktion F (x) von x von beliebigem Grade

,,der Coëﬃcienten des höchstens Gleid = 1 ist, und alle folgenden Coëﬃ-

,,cienten ganze (reelle, complexe) Zahlen sind, in welchen eine gewisse

,,(reelle resp. complexe) Primzahl m aufgeht, wenn ferner der letzte

,,Coëﬃcient = εm ist, wo ε eine nicht durch m teilbare Zahl vorstellt:

,,so ist es unmöglich F(x) auf die Form

+ a

µ−1

+ . . . . + a

)(x

+ b

ν−1

+ . . . . + b

)

,,zu bringen, wo µ und ν ≥ 1, µ + ν = dem Grad von F (x), und alle

,,a und b (reelle resp. complexe) ganze Zahl sind; und die Gleichung

,,F (x) = 0 is demnach irreductibel.”

If in a polynomial F (x) of x of arbitrary degree the coeﬃcient of

the highest term is = 1, and all following coeﬃcients are integers (real

or complex), in which a certain (real resp. complex) prime number m

appears, if further the last coeﬃcient is = εm, where ε represents a

number not divisible by m: then it is impossible to bring F (x) into the

form

+ a

µ−1

+ . . . . + a

)(x

+ b

ν−1

+ . . . . + b

)

where µ and ν ≥ 1, µ + ν = the degree of F (x), and all a and b are

(real resp. complex) integers; and the equation F (x) = 0 is accordingly

irreducible.

There are two standard proofs of the Eisenstein criterion. One proof (due to Eisenstein) works

by studying which coeﬃcients of the factors are divisible by the prime. The other proof (due to

Schönemann) reduces modulo p and uses unique factorization in F

[x].

Normat 2/2009 David A. Cox 69

After giving the proof (which works over any unique factorization domain), Eisen-

stein applies his criterion to the equation W = 0 that arises from division of the

lemniscate and also to our friend x

p−1

+ ··· + 1. Eisenstein’s proof that the latter

is irreducible is essentially identical to the one sketched on the ﬁrst page of this

article.

Eisenstein’s paper is the ﬁrst appearance of this classic proof of the irreducibil-

ity of x

p−1

+ ··· + 1. Eisenstein is clearly pleased to have found such a splendid

argument:

. . . Dies giebt also, wenn man will, einen neuen and höchst einfachen

Beweis der Irreducibilität der Gleichung x

p−1

+ x

p−2

+ . . . . + x + 1 = 0;

und zwar setzt dieser Beweis in Unterschiede mit früheren

∗∗

) nicht die

Kenntiss der Wurzeln und ihrer gegenseitigen Abhängigkeit voraus.

∗∗

) Ausser dem Beweise von Gauss ist mir nur der von Kronecker im

29ten Bande dieses Journals Seite 280 bekannt.

. . . This thus gives, if you will, a new and most highly simple proof

of the irreducibility of the equation x

p−1

+ x

p−2

+ . . . . + x + 1 = 0;

and in constrast with earlier ones

∗∗

), this proof does not presuppose

knowledge of the roots and the relations among them.

∗∗

) Besides the proof of Gauss, only that of Kronecker in volume 29

of this journal, page 280, is known to me.

We know about Gauss’s proof, and Kronecker’s proof [18, Vol. I, pp. 1–4] from

1845 is simpler than Gauss’s but still uses the explicit relations among the roots.

But notice what footnote does not mention: Schönemann’s two proofs of the irre-

ducibility of x

p−1

+ ··· + 1 given in his papers of 1845 and 1846. Yet Eisenstein’s

paper appears in the same journal in 1850!

Schönemann Complains

Eisenstein’s paper, with the oﬀending footnote, appeared in volume 39 of Crelle’s

journal. In volume 40, Schönemann published a Notiz [26], which began by describ-

ing two theorems from Eisenstein’s paper:

• (ersten/ﬁrst): The Eisenstein criterion for real primes (in Z) and complex

primes (in Z[i]).

• (letzterem/last): The irreducibility of the cyclotomic polynomial x

p−1

+···+1,

proved using the Eisenstein criterion.

Then Schönemann goes on to say:

. . . Da Herr Eisenstein ausdrücklich bemerkt, dass ihm von letzterem

Satze nur der Beweis von Gauss und von Kronecker bekannt ist, so

sehe ich mich veranlasst, daran zu erinnern, das ich bereits im Bande

70 David A. Cox Normat 2/2009

31 dieses Journals §. 6, in meiner Abhandlung ,,Grundzüge einer allge-

meinen Theorie der höhern Congruenzen etc.“ den ersten Satz für reelle

Primzahlen beweisen und ach den folgenden aus demselben abgeleitet

habe und das ferner die von Herrn etc. Eisenstein angewandete Meth-

ode nicht wesentlich von der meinigen verschieden is. Von dem letzteren

Satze habe ich übrigens noch einen ganz verschiedenen Beweise im er-

sten Theile und §. 50 derselben Abhandlung gegeben.

. . . Since Eisenstein expressly noted, that for the last theorem he only

knew the proofs of Gauss and Kronecker, I am led to recall that in §.

6 of my paper ,,Foundations of a general theory of higher congruences

etc.” in volume 31 of this journal, I proved the ﬁrst theorem for real

primes and deduced the last from the ﬁrst, and also the method used

by Eisenstein is not signiﬁcantly diﬀerent from mine. For the last

theorem, I in addition even gave an entirely diﬀerent proof in §. 50 of

the ﬁrst part of the paper.

It seems clear that Eisenstein messed up by not citing Schönemann. However,

there are some complications and confusions. First, Schönemann refers to §6 of his

Grundzüge .. . paper in volume 31 of Crelle’s journal, yet his irreducibility criterion

and its application to x

p−1

+ ··· + 1 are in §61 of the second part of his paper,

which appeared in volume 32. The “§. 6” in his Notiz should have been “§. 61.”

This explains part of the reason I had trouble ﬁnding Schönemann’s criterion—I

was looking in the wrong section!

But there was also confusion on Eisenstein’s side as well. As already noted, Eisen-

stein’s study of the division equations of the lemniscate was published in a two-part

paper in Crelle’s journal. The footnote quoted above appeared in the ﬁrst part, in

issue II of volume 39. The second part of the paper, Über einige allgemeine Eigen-

schaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt,

nebst Anwendungen derselben auf die Zahlentheorie (On some general properties of

equations that depend on the division of the lemniscate, together with applications

to number theory) [10, pp. 555—619], appeared in issue III of the same volume.

This paper included an explicit reference to Schönemann’s ﬁrst proof of the irre-

ducibility of x

p−1

+···+ 1 (the one from §50 of Schönemann’s paper in volume 31).

Yet somehow this proof was unknown to Eisenstein when he wrote the ﬁrst part

of his paper. One can speculate on why this happened, but we will never know for

sure.

Conclusion

We are now at the end of the amazing story of how Schönemann and Eisenstein

independently discovered their criteria. Given that Schönemann discovered it ﬁrst,

the name “Schönemann-Eisenstein criterion” used by Dorwart is the most histori-

cally accurate. However, since most people use the Eisenstein’s version, the name

“Eisenstein-Schönemann criterion” is also reasonable. However, in my view, the

name “Eisenstein criterion” does not do justice to Schönemann.

In the quote from Section VII of Disquisitiones, Gauss acknowledged two items

of unﬁnished business: the extension from circular to transcendental functions such

Normat 2/2009 David A. Cox 71

as Abel’s lemniscatic function ϕ, and the study of higher congruences. Both led to

major areas of modern mathematics (elliptic curves and complex multiplication in

the ﬁrst case, p-adic numbers and local methods in number theory in the second),

and both led to the Schönemann-Eisenstein criterion. Schönemann followed higher

congruences to Hensel’s Lemma to a question about irreducibility modulo p

: his

criterion appears in a completely natural way. Eisenstein followed Abel’s work the

lemniscate and considered the coeﬃcients of the resulting division polynomials: his

criterion appears in a completely natural way, completely diﬀerent from the context

considered by Schönemann. Yet both have their origin in the same paragraph in

Disquisitiones. As I said, it is an amazing story.

Acknowledgements

The English translations of the ﬁrst two Gauss quotes are from the English version

of [13]. For the third Gauss quote and the ﬁrst two Schönemann quotes, I used [11].

I would also like to thank Annemarie and Günter Frei for help in understanding the

salutation in Eisenstein’s letter to Gauss. Thanks also to Michael Filaseta for his

help in pointing me to the right place in Schönemann’s papers and to David Leep

for bringing Dorrie’s book [8] to my attention. I am also grateful to the referee for

several useful suggestions.

I should also mention that the papers from Crelle’s journal quoted in this article

are available electronically through the Göttinger Digitalisierungzentrum at the

web site

http://gdz.sub.uni-goettingen.de/dms/load/toc/?IDDOC=238618

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