Eisenstein Criterion: Proof and Discovery

WHY EISENSTEIN PROVED THE EISENSTEIN CRITERION AND WHY SCHÖNEMANN DISCOVERED IT FIRST The Eisenstein irreducibility critierion is part of the training of every mathematician. 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 SchoenemannEisenstein theorem: If, in the integral polynomial a0 xn + a1 xn−1 + · · · + an , all of the coefficients except a0 are divisible by a prime p, but an is not divisible by p2 , 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 irreducibility of the so-called “cyclotomic polynomial” xp − 1 f (x) = = xp−1 + xp−2 + · · · + 1, x−1 where p is prime. If, instead of f (x), we consider f (x + 1), where   (x + 1)p − 1 p p−2 p−1 f (x + 1) = =x + x + · · · + p, (x + 1) − 1 1 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.”1 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 1This edition included a reference to Schönemann that was dropped in the 1937 second edition. 1 2 will be far from complete. I will instead focus on specific highlights to trace the development 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 xp−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 formae xm + Axm−1 + Bxm−2 + Cxm−3 . . . .. + N . . . . . . . . . . . . . . . . . (P ) xµ + axµ−1 + bxµ−2 + cxµ−3 . . . .. + n . . . . . . . . . . . . . . . . . (Q) omnes sunt rationales, neque vero omnes integri, productumque ex (P ) et (Q) = xm+µ + Axm+µ−1 + Bxm+µ−2 + etc. + Z omnes coëfficientes A, B . . . . Z integri esse nequeunt. 42. If the coefficients A, B, C . . . . N ; a, b, c . . . . n of two functions of the form xm + Axm−1 + Bxm−2 + Cxm−3 . . . .. + N . . . . . . . . . . . . . . . . . (P ) xµ + axµ−1 + bxµ−2 + cxµ−3 . . . .. + n . . . . . . . . . . . . . . . . . (Q) are all rational and not all integers, and if the product of (P ) and (Q) = xm+µ + Axm+µ−1 + Bxm+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 xn −1 = 0 is solvable by radicals. The modern approach to solvability by radicals allows the introduction of arbitrary roots of unity, which implies that xn − 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(e2πi/p ) when p is prime. This has degree p − 1 by the irreducibility of xp−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 circulares, sed pari successu ad multas functiones transscendentes applicari possunt, e.g. ad eas, quae R alias dx ad integrali √(1−x 4 ) pendent, praetereaque etiam ad varia congrueniarum genera: sed quoniam de illis functionibus transscendentibus amplum opus peculiare paramus, de congruentiis autem in continuatione disquitionum arithmeticarum copiose 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 3 as to other transcendental functions, e.g. to those which depend on the integral R well dx √(1−x4 ) 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 length in the continuation of these Disquisitiones, we have decided to consider only circular functions here. In this quote, the reference to functions is clear. But what about transcendental functions R circular dx that depend on the integral √(1−x 4 ) ? Here, any 19th century mathematician would immediately R 1 dx think of the lemniscate r2 = cos 2θ, whose arc length is 4 0 √(1−x 4 ) . This integral and its relation to the lemniscate were discovered 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 Fp [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) = pn − P n pa + P n p ab − P n p abc etc. P n P n where the sum p a (resp. p ab ) 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 congruences. 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 pluriumve dimensionum resolvi possit, unde radices quasi imaginariae illi attribui possint. Revera, si simili licentia, quam recentiores mathematici usurparunt, uti talesque quantitates imaginarias introducere voluissemus, omnes nostras disquisitiones sequentes incomparabiter contrahere licuisset; . . . . . . but nothing prevents us from decomposing ξ, nevertheless, into factors 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 liberty some more recent mathematicians have taken; . . . 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. 4 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 y 2 = (1 − c2 x2 )(1 + e2 x2 ), (1) and his elliptic functions were inverse functions to the elliptic integral Z Z dx dx p = . y (1 − c2 x2 )(1 + e2 x2 ) 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 Z Z p p (2) dy/ 1 − y 4 = m dx/ 1 − x4 .” 0 0 In modern language, we are talking about division points on elliptic curves, and the “algebraic integral” produces a polynomial Pm of degree m2 whose solutions give (roughly speaking) the mdivision points on the associated elliptic curve defined by (1). We will say more about the polynomial Pm and the equation (2) when we discuss Eisenstein. For Abel and his contemporaries, a central question was whether polynomial equations such as Pm (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 journal2 in 1827 and 1828, Abel considers the equation P2n+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 P2n+1 = 0 est reduite á celle d’une seule équation du degré 2n + 2; mais en général cette équation ne paraı̂t 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 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 2n ou 2n + 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 P2n+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 completely in many particular cases, for √ can solve it √ example, when e = c, e = c 3, 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: 2The Journal für die reine und angewandte Mathematik, founded by August Leopold Crelle in 1826. 5 “One can divide the entire circumference of the lemniscate into m parts by ruler and compass only, if m is of the form 2n or 2n + 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 radicals 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. Again we are in the presence of remarkably rich mathematics. Abel thought deeply about why his equations P2n+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 algébriquement [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 xn − 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 xn − 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 xn − 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 xn−1 + · · · + x + 1 = 0. Define the polynomial θ(x) = xα , where α is a primitive root modulo n. Then the roots are given by 2 3 n−2 x, θ(x) = xα , θ2 (x) = xα , θ3 (x) = xα , . . . , θn−2 (x) = xα , where θn−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 rationnellement au moyen de l’une d’elles, que nous désignerons par x; si de plus, en désignant par θx, θ1 x deux 6 autres racines quelquonques, on a θθ1 x = θ1 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 themselves in such a way, that all of the roots can be rationally expressed in terms of one of them, which we designate by x; if in addition, designating by θx, θ1 x two other arbitrary roots, one has θθ1 x = θ1 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 σi of the Galois group Gal(L/K), there is a polynomial θi (x) ∈ K[x] such that σi (α) = θi (α). Then one easily computes that σi σj (α) = θj (θi (α)). The switch of indices is correct—you should check why. Since σi is determined by its value on α, σi σj = σj σi ⇐⇒ θj (θi (α)) = θi (θj (α)). Since the θi (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 fonctions 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 functions 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), . . . , θn−1 (x), x = θn (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. • In 1870, Jordan [17, p. 287] defined f (x) = 0 to be “Abelian” in terms of its Galois group: 7 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 contains 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.3 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 leminiscate, 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, F x = 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 + a1 i + a2 i2 + · · · + aν−1 iν−1 , where ν = deg(F ) and a, a1 , . . . , aν−1 are integers modulo p, form a finite field with pν 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. Schönemann. Unlike the other people mentioned so far, Theodor Schönemann is not a familiar 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 became 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 3In 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]. 8 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 höhern 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 polynomials 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 polynomials 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: • The “allgemeine Form eines kleines Restes” (“general form of a smallest remainder”) is a0 αn−1 + a1 αn−2 + · · · + an−1 , where ai ∈ {0, . . . , p − 1}. This gives the finite field Fpn . n • The elements of Fpn are the roots of xp − x ≡ 0 mod (p, α). n−1 • f (x) ≡ (x − α)(x − αp ) · · · (x − αp ) mod (p, α). Thus Fpn 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 Φp (x) = xp−1 + · · · + x + 1. We will give the proof in modern notation. Pick a prime ℓ 6= p and consider the prime factorization Φp (x) ≡ f1 (x) · · · fr (x) mod ℓ. where the fi are irreducible modulo ℓ. Standard properties of finite fields imply that deg(fi ) = the minimum n such that F∗ℓn has an element of order p (3) = the minimum n such that ℓn ≡ 1 mod p = the order of the congruence class of ℓ in (Z/pZ)∗ . 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 ℓ. 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 ℓ. This implies that Φp (x) is irreducible modulo ℓ and hence irreducible over Z. Then Φp (x) is irreducible over Q by Gauss’s Lemma. 9 This proof is simpler than Gauss’s, though it does require knowledge of finite fields. The use of the auxiliary prime ℓ 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 factorization of polynomials modulo pm , 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 Modul keinen gemeinsaftlichen Divisor haben: so ist dieser Ausdruck auch nach dem Modul pm , aber nur auf einer Weise, in zwei Factoren zerlegbar, welche jenen beiden ersten nach dem modul p congruent sind. 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 pm , in a unique manner, into two factors, which are congruent to those first two factors modulo p.4 (Here, “einfacher Ausdruck von x” means a monic polynomial of x.) As a consequence, when an irreducible polynomial modulo pm 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 pm irreductibel sei, oder nicht. Problem. To investigate, whether the power of irreducible polynomial modulo p is or is not irreducible modulo pm . An especially simple example is (x − a)n , and for a polynomial congruent to (x − a)n modulo p, the first place to check for irreducibility is modulo p2 . Here is Schönemann’s answer: . . . man darf daher den Satz aussprechen: dass (x − a)n + pFx nach dem Modul p2 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)n + pFx is irreducible modulo p2 , 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.5 Since x − a divides F (x) modulo p if and only if F (a) ≡ 0 mod p, we can state Schönemann’s result as follows. 4The 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) ∈ Zp [x], a solution of f (x) ≡ 0 mod p of multiplicity one lifts to a solution of f (x) = 0 in Zp . 5For example, let F (x) = x3 − p2 x + 1. Then x2 + pF (x) = (px + 1)(x2 − p2 x + p), yet x does not divide F (x) modulo p. 10 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)n + pF (x), If F (a) 6≡ 0 mod p, then f (x) is irreducible modulo p2 . The proof is not difficult (assume (x − a)n + pF (x) factors modulo p2 , reduce modulo p and use unique factorization in Fp [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) = a0 xn + a1 xn−1 + · · · + an satisfies the hypothesis of the Eisentstein criterion. Multiplying by a suitable integer, we may assume a0 ≡ 1 mod p. This allows us to write f (x) = xn + pF (x). Note also that F (0) 6≡ 0 mod p since p2 does not divide an . Then f (x) is irreducible modulo p2 by Schönemann’s criterion. This implies irreducibility over Z and hence (via Gauss’s Lemma) over Q. As you might expect, Schönemann immediately applies his irreducibility criterion to a familiar polynomial: xn − 1 Wenden wir das erhaltene Resultat auf der Ausdruck an, wo n eine x−1 n n Primzahl bedeutet. Es ist für diesen Fall x − 1 ≡ (x − 1) (mod. n), und man erhält also xn − 1 = xn−1 + xn−2 + · · · · + x + 1 = (x − 1)n−1 + nF x. x−1 Für x = 1 erhält man n = nF (1) und daher F (1) = 1, und nicht ≡ 0 (mod. n). xn − 1 Hieraus folgt, dass nach dem Modul n2 stets irreductibel ist, wenn x−1 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.) xn − 1 , where n denotes x−1 n n a prime number. In this case x − 1 ≡ (x − 1) (mod. n), and one thus obtains Let us apply the result just obtained to the polynomial xn − 1 = xn−1 + xn−2 + · · · · + x + 1 = (x − 1)n−1 + nF x. x−1 For x = 1 one obtains n = nF (1) and thus F (1) = 1, and not ≡ 0 (mod. n). xn − 1 From this, it follows that is always irreducible modulo n2 , if n x−1 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 xn−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.) 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, 11 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 p2 . 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 p2 : 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 p2 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 pk 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 multiplices resolutio ad congruentias secundum modulum linearem reducitur. Beginning to uncover principles, by which the resolution of congruences according to multiple moduli is reduced to congruences according to linear moduli. Here, “resolution of congruences according to multiple moduli” means factoring polynomials modulo pk , 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)m 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 generales 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. 12 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 depends 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 modulus) [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 is a finite field with pn 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 ∗ , µ1 , . . . , µs (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 µi + µj = µj + µi ; µi µj = µj µi ; (µi + µj )µk = µi µk + µj µk ; 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 language 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 following 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. 13 Moore was writing for a mathematically sophisticated audience, but he didn’t assume 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 undergraduates. 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 Z r dr √ (4) r = ϕ(s) ⇐⇒ s = . 1 − r4 0 (We follow the 19th century practice of using the same letter for the variable and limit of integration.) R 1 dr In this equation, 0 ≤ r ≤ 1 corresponds to 0 ≤ s ≤ ̟ = 0 √1−r . Then define ϕ for s ≥ 0 by 4 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 on the next page. An arc length calculation shows that s and the radius r are related by the equation s −1 r 1 Figure 1. Arc length on the lemniscate r = ϕ(s) (see [4, §15.2]). In particular, ̟ is one-fourth of the total arc length of the lemniscate, so that ϕ(̟) = 1 and ϕ(2̟) = 0. Hence, for any positive integer m, r = ϕ(k · 2̟/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.6 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 (5) y= A0 x + A1 x5 + · · · + A(N (m)−1)/4 xN (m) U (x) = V (x) 1 + B1 x4 + · · · + B(N (m)−1)/4 xN (m)−1 where N (m) = a2 + b2 . See [4, Thm. 15.4.4] for a proof. 6Gauss followed the same path in 1797, though he never published his findings. See [3] for more details. 14 When m is an ordinary odd integer, we know that r = ϕ(k · 2̟/m) gives m-division points on the lemniscate. Setting y = ϕ(m · (k · 2̟/m)) = ϕ(k · 2̟) = 0 and x = ϕ(k · 2̟/m) = r into (5), we see that U (r) , hence U (r) = 0. V (r) 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 P2n+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].7 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 Z y Z x dy dx p √ =m . 4 1 − x4 1−y 0 0 0= 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: U Wenn m = a + bi eine ungerade complexe Zahl, p deren Norm und y = = V 5 p A0 x + A1 x + · · · · +A(p−1)/4 x das algebraische Integral der Gleichung 1 + B1 x4 + · · · · +B(p−1)/4 xp−1 Z Z p p dy/ 1 − y 4 = m dx/ 1 − x4 0 0 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 = U = V A0 x + A1 x5 + · · · · +A(p−1)/4 xp is the algebraic integral of the equation 1 + B1 x4 + · · · · +B(p−1)/4 xp−1 Z Z p p 4 dy/ 1 − y = m dx/ 1 − x4 , 0 0 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); 7For 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. 15 (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 = a2 + b2 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 1 U (x) = A0 + A1 x4 + · · · + A(p−1)/4 xp−1 . x 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 A0 , . . . , 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 A0 is m, which is not divisible by m2 . 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 Funktionen von x mit ganzen complexen Coefficienten, und deren Grade < p− 1 sind. Es sei W = P Q; 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. W (x) = . . . 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 Eminence8 used in the real number theory (Disq. Section I). 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.9 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 lemniscate; and • appeared in a letter to Gauss. 8The 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. 9There 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 Fp [x]. 16 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 Irreductibilität 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 Coëfficienten ganze (reelle, com,,plexe) 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 teil,,bare Zahl vorstellt: so ist es unmöglich F (x) auf die Form (xµ + a1 xµ−1 + . . . . + aµ )(xν + b1 xν−1 + . . . . + bν ) ,,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 (xµ + a1 xµ−1 + . . . . + aµ )(xν + b1 xν−1 + . . . . + bν ) 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. After giving the proof (which works over any unique factorization domain), Eisenstein applies his criterion to the equation W = 0 that arises from division of the lemniscate and also to our friend xp−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 irreducibility of xp−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 Irreductibilität der Gleichung xp−1 + xp−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 xp−1 + xp−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 17 mention: Schönemann’s two proofs of the irreducibility of xp−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 describing 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 xp−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 31 dieses Journals §. 6, in meiner Abhandlung ,,Grundzüge einer allgemeinen 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 Methode nicht wesentlich von der meinigen verschieden is. Von dem letzteren Satze habe ich übrigens noch einen ganz verschiedenen Beweise im ersten 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 xp−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, Eisenstein’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 Eigenschaften 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 irreducibility of xp−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 historically accurate. However, 18 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 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 p2 : 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 References [1] N. H. Abel, Oeuvres complètes de Niels Henrik Abel, Volume I, Edited by L. Sylow and S. Lie, Grøndahl & Søn, Christiana, 1881. [2] Allgemeine Deutsche Biographie, Duncker & Humblot, Leipzig, 1875–1912. Available electronically at http://www.deutsche-biographie.de/∼ndb/adb index.html http://de.wikisource.org/wiki/Allgemeine Deutsche Biographie [3] D. A. Cox, The arithmetic-geometric mean of Gauss, Enseign. Math. 30 (1984), 275–330. Reprinted in Pi: A Source Book, (L. Berggren, J. Borwein and P. Borwein, eds), third edition, Springer, 2003, 481–536. [4] D. A. Cox, Galois Theory, Wiley, Hoboken, NJ, 2004. [5] D. A. Cox, Primes of the Form x2 + ny 2 , Wiley, Hoboken, NJ, 1989. [6] R. 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Math. 31 (1845), 269–325. T. Schönemann, Von denjenigen Moduln, welche Potenzen von Primzahlen sind, J. Reine Angew. Math. 32 (1846), 93–105. T. Schönemann, Notiz, J. Reine Angew. Math. 40 (1850), 188. J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259– 331. B. L. van der Waerden, Moderne algebra, Springer, Berlin, 1930. H. Weber, Lehrbuch der Algebra, second edition, Vieweg, Braunschwieg, 1898–1908. (Reprint by Chelsea, New York and AMS Chelsea, Providence, RI, 1961.) Department of Mathematics, Amherst College, Amherst, MA 01002, USA E-mail address: dac@cs.amherst.edu