§12.1 A GROUP IS DEFINED RATHER simply by means of just four axioms: closure, associativity, identity, and inverse (§11.4). This very simplicity gives groups a wide scope, just as there is more variety encompassed by the simple description “four-legged creature” than by a lengthier description like “four-legged creature with tusks and trunk.” It is the very simplicity of the group definition that allows groups to be applied to things far from the realm of mere numbers. It also gives groups the leeway to possess a complicated and interesting inner structure, with normal subgroup as the key concept.

A field (§FT.2) is a more complex object, needing 10 axioms for its definition. Instead of just one basic rule of combination, it has two: addition and multiplication. (Subtraction and division are really just addition and multiplication of inverses: 8 − 3 is the same as 8 + (−3).) This greater complexity keeps the concept of “field” more closely tied to ordinary numbers. It also, paradoxically, restricts their possibilities for having interesting inner structure.

There is another mathematical object much studied by modern algebra: the ring. A ring is more complicated than a group but less complicated than a field, so while it is not as wide ranging in its applications as the group, it can roam farther away from ordinary-number

applications than a field can. Like a group, a ring can have an interesting inner structure. The key concept here is the ideal. I shall explain that at some length in this chapter and the next.

In fact, as often happens with intermediate notions, the ring concept offers mathematicians something of the best from both worlds. It stands, for example, at the center of modern algebraic geometry, source of the deepest and most challenging ideas in modern algebra. It was, however, an unusually long time before the full power of the ring concept came to be appreciated.

It used to be said that ring theory all began with Fermat’s Last Theorem. That turns out to have been a mistake. Fermat’s Last Theorem is a good hook on which to hang the beginnings of ring theory, though, so that is where I shall start.

§12.2 I mentioned Pierre de Fermat and his last theorem in §2.6. Scribbled in the margin of Fermat’s copy of Diophantus’s Arithmetica around 1637, the theorem asserts that the equation

has no solutions in positive whole numbers x, y, z, and n when n is greater than 2. Fermat himself, in 1659, sketched a proof for the case n = 4, and Gauss provided the complete proof much later. Euler offered a proof for the case n = 3 in 1753.

There was then no real progress for half a century until the French mathematician Sophie Germain—second of the three great female mathematicians in this book (giving Hypatia the benefit of the doubt)—proved that Fermat’s Last Theorem is true for a large general class of integers x, y, z, and n. It would be too much of a digression to explain which quartets of integers fall into that class. Suffice it to say that the next steps in progress on the theorem built on Sophie Germain’s result.

The French mathematician Adrien-Marie Legendre (who was 72 at the time) and the German (despite his French name) Lejeune

Dirichlet separately proved the n = 5 case of the theorem in 1825. By this time everyone understood that the only real challenge was to prove the theorem for prime numbers n, so the next target was n = 7. Another Frenchman, Gabriel Lamé, solved that one in 1839. At this point, however, the story took a new turn.

§12.3 Just to remind you, the set of integers consists of all the positive whole numbers, all the negative whole numbers, and zero.

(This list heads off to infinity at both left and right.) Now, is not a field. You can add, subtract, and multiply freely, and the answer will still be in . However, division works only sometimes. If you divide −12 by 4, the answer is in . If you divide −12 by 7, however, the answer is not in . Since you can’t divide freely without straying outside the bounds of , it is not a field.

is, though, sufficiently interesting and important to be worth the attention of mathematicians in its own right. Even if division can’t be made to work reliably, you can do a great deal with addition, subtraction, and multiplication. You can, for example, explore issues of factorization and primality (that is, the quality of being a prime number).

Furthermore, there are other kinds of mathematical objects that resemble in allowing addition, subtraction, and multiplication freely but throwing up barriers to division. This is so, for example, with polynomials. Given two polynomials, say x⁵ − x and 2x² + 3x + 1, we can add them to get another polynomial (answer: x⁵ + 2x² + 2x + 1), or subtract them (answer: x⁵ − 2x² − 4x −1), or multiply them (answer: 2x⁷ + 3x⁶ + x⁵ − 2x³ − 3x² − x). We can’t necessarily divide them to get another polynomial, though. In this particular case we certainly can’t, though in other cases we sometimes can: (2x² + 3x + 1) ÷ (x + 1) = (2x + 1). Just like integers!¹¹⁵

This kind of mathematical object, in which the first three pocket calculator rules work reliably, but the fourth doesn’t, is called a ring.¹¹⁶ Now you can see what I meant by saying that rings stand between groups and fields in the order of things. “Field” is more tightly defined than “ring,” with a full capability for division; “group” is defined more loosely, with only one way to combine elements. Look at the four axioms in §11.4 defining an abstract group. A field needs 10 axioms, a ring only 6.

In the course of some work in number theory, Gauss had discovered a new kind of ring, one involving complex numbers. Gauss did not have the word “ring” available to him—it did not show up until a hundred years later—or even the abstract concept, but a ring is what he discovered nonetheless. This was the ring of what we now call Gaussian integers, complex numbers like −17 + 22i, whose real and imaginary parts both belong to . You can develop—Gauss did develop—an entire integer-like arithmetic with these “complex integers.”

This arithmetic is not at all straightforward. Speaking very generally, rings are even less division-friendly than they at first appear to be. You can get some division going in without too much difficulty and develop the theory of prime numbers and factorization familiar from ordinary arithmetic. The fact that the negative numbers are included in adds only a few small and inconsequential wrinkles. Getting a good theory of primes and factorization up and running in other rings is usually much more difficult. It can be done with Gauss’s family. It can also be done with a different family that Euler used in his work on the n = 3 case of Fermat’s Last Theorem, a family that is slightly “bigger” than Gauss’s, allowing as well as integers. When you get deeper into these kinds of rings, though, ugly things start to happen.

The ugliest thing is that unique factorization breaks down. In the ring , any integer can be expressed as a product of a unit and a set of primes in just one way. (A unit, in ring theory, is a number that divides into 1. has two units, 1 and −1. Gauss’s ring of complex inte-

gers has four: 1, −1, i, and −i.¹¹⁷) The integer −28, for instance, factorizes to −1 × 2 × 2 × 7. Other than by just rearranging the order of factors, you can’t get it to factorize any differently. The ring is blessed with the property of unique factorization.

On the other hand, consider the ring of numbers where a and b are ordinary integers. In this ring, the number 6 can be factorized two ways, as 2 × 3 and also as . This is alarming, because all four of those factors are prime numbers in this ring (defined to mean they have no factors except themselves and units). Unique factorization has broken down.

§12.4 This unhappy state of affairs led to the great debacle of 1847. By this point the French Academy was offering a gold medal and a purse of 3,000 francs for a proof of Fermat’s Last Theorem. Following his successful assault on the case n = 7, Gabriel Lamé announced to a meeting of the Academy, on March 1 of that year, that he was close to completing a general proof of the theorem, a proof for all values of n. He added that his idea for the proof had emerged from a conversation he had had with Joseph Liouville some months before.

When Lamé had finished, Liouville himself stood up and poured cold water on Lamé’s method. He pointed out that it was, in the first place, hardly original. In the second place, it depended on unique factorization in certain complex number rings, and this could not be depended on.

Cauchy then took the floor. He supported Lamé, said that Lamé’s method might well deliver a proof, and revealed that he himself had been working along the same lines and might soon have a proof of his own.

This meeting of the Academy was naturally followed by some weeks of frantic activity on Fermat’s Last Theorem, not only by Lamé and Cauchy but also by others who had been attracted to the cash prize.

Then, 12 weeks later, before either Lamé or Cauchy could announce their completed proofs, Liouville read a letter to the Academy. The letter was from a German mathematician named Eduard Kummer, who had been following the Paris proceedings on the mathematical grapevine. Kummer pointed out that unique factorization would indeed break down under the approaches taken by Lamé and Cauchy, that he himself had proved this three years earlier (though he had published the proof in a very obscure journal), but that the situation could be recovered to some extent by the use of a concept he had published the previous year, the concept of an ideal factor.

It used to be said (E. T. Bell says it in Men of Mathematics) that Kummer had developed this new concept in the course of work he himself had been doing on Fermat’s Last Theorem. Modern scholars, however, believe that Kummer had in fact done no work on the theorem until after discovering these ideal factors. Only then, and alerted to the fuss in Paris, had he tackled the theorem.

Within a few weeks after Liouville’s reading of his letter, Kummer sent in a paper to the Berlin Academy proving Fermat’s Last Theorem for a large class of prime numbers, the so-called regular primes.¹¹⁸ His proof used the ideal factors he had discovered. This was the last really important advance in the attack on Fermat’s Last Theorem for over a century. The theorem was finally proved by Andrew Wiles in 1994.

But what were these “ideal factors”? It is not easy to explain. Historians of mathematics do not usually bother to explain it, in fact, because Kummer’s ideal factors were soon superseded by the larger, more general, and more powerful concept of a ideal, which is not a number but a ring of numbers. I think this is a bit unfair to Kummer, so here is an outline of his concept.

Kummer was working with cyclotomic integers, a concept I shall pause to explain very briefly. The reader may recall the word “cyclotomic” from my primer on roots of unity, §RU.2. When this word shows up in math, you are never far away from the roots of unity. Suppose p is some prime number. What are the pth roots of unity? Well, the

number 1 is of course a pth root of unity. The others are scattered evenly round the unit circle in the complex plane, as in Figure RU-1. If we call the first one (proceeding counterclockwise from 1) a, then the others are a², a³, a⁴, …, a^p−1.

A cyclotomic integer is a complex number having the form

where all the capital letter coefficients are ordinary integers in and α is a pth root of unity. If p is 3, for example, then the roots of unity are our old pals 1, ω, and ω², the latter two being roots of the quadratic equation 1 + ω + ω² = 0 (§RU.3). An example of a cyclotomic integer for the case p = 3 would be 7 − 15ω + 2ω². Note that this is a perfectly ordinary complex number, . I have just chosen to write it in terms of 1, ω, and ω².

These cyclotomic integers have some weird and wonderful properties. Sticking with the case p = 3, for example, from 1 + ω + ω² = 0, it follows that for any integer n, n + nω + nω² = 0. Since adding zero to a number leaves it unchanged, I can add the left-hand side to 7 − 15ω + 2ω², giving (n + 7) + (n − 15)ω + (n + 2)ω², without changing it, a fact you can easily confirm by substituting the actual values of ω and ω². Just as , , , , and an infinity of other fractions all represent the same rational number, so that second form of my cyclotomic integer, for any value of n at all, will always represent the same cyclotomic integer.

Well, Kummer’s work concerned the factorization of these cyclotomic integers. This turns out to be a deep and knotty issue. As you might guess, the problem of unique factorization breaking down soon arises. (Though not very soon: It first happens when p = 23. This is one reason the theory of ideal factors is hard to illustrate.) This was the particular problem Kummer tackled. He solved it by tightening the ordinary definition of prime number to make it more suitable for cyclotomic integers. Kummer then built up his ideal factors from these “true primes” to get a full theory of factorization for cyclotomic integers.

Out of all this, Kummer was able to prove his great result showing that Fermat’s Last Theorem is true for regular primes. This was, though, a particular and local application. Before the full power of ring theory could be revealed, a higher level of generalization had to be attained. This higher level was reached by the following generation of mathematicians.

§12.5 Eduard Kummer’s 1847 letter to the French Academy had significance beyond the merely mathematical. Kummer was 37 at the time, working as a professor at the University of Breslau in Prussia. The unification of Germany was still 20 years in the future, but national feeling was strong, and the German people as a people, if not yet a nation, were the great rising force in European culture. Resentment of France for the indignities she had inflicted on Germany during the Napoleonic wars still ran strong after 40 years.¹¹⁹

Kummer felt this resentment keenly. His father, a physician in the little town of Sorau, 100 miles southeast of Berlin,¹²⁰ had died when Eduard was three years old, from typhus carried into the district by the remnants of Napoleon’s Grand Army on its retreat from Russia. As a result, Kummer had grown up in dire poverty. Though he seems to have been a pleasant enough fellow and a gifted and popular teacher, it is hard not to suspect that Kummer must have felt a twinge of satisfaction at showing the French Academy who was boss.

The defeats and humiliations inflicted on the Germans by Napoleon had had a larger consequence, too. They had spurred Prussia, with the lesser German states following close behind, to overhaul her systems of education and of technical and teacher training. The harvest from this, and from the prestige and example of the mighty Gauss, was a fine crop of first-class German mathematicians at midcentury: Dirichlet, Kummer, Helmholtz, Kronecker, Eisenstein, Riemann, Dedekind, Clebsch.

By the time national unification arrived in 1866, Germany could even boast two great centers of mathematical excellence, Berlin and

Göttingen, each with its own distinctive style. The Berliners favored purity, density, and rigor;¹²¹ Göttingen mathematics was more imaginative and geometrical—a sort of Rome/Athens contrast. Weierstrass and Riemann exemplify the two styles. Weierstrass, of the Berlin school, could not blow his nose without offering a meticulous eight-page proof of the event’s necessity. Riemann, on the other hand, threw out astonishing visions of functions roaming wildly over the complex plane, of curved spaces, and of self-intersecting surfaces, pausing occasionally to drop in a hurried proof when protocol demanded it.

And while this was happening, French mathematics had gone into a decline. That is to speak relatively: A nation that could boast a Liouville, an Hermite, a Bertrand, a Mathieu, and a Jordan was not starving for mathematical talent. Paris’s mathematical high glory days were behind her, though. Cauchy died just two years after Gauss, but Cauchy’s death marked the end of a great era of mathematical excellence in France, while Gauss’s occurred as German mathematics was rising fast.

§12.6 Richard Dedekind was of the best in that midcentury crop of German mathematicians. A serene and self-contained man who cared about nothing very much except mathematics, Dedekind lived a nearly eventless life, most of it as a college teacher in his (and Gauss’s) hometown of Brunswick.

Dedekind’s contribution to algebra was threefold. First, he gave us the concept of an ideal. Second, he, with Heinrich Weber, opened up the theory of function fields—the theory of which I gave a very brief hint at the end of my primer on fields. (There are more details on this in §13.8.) Third, Dedekind began the process of axiomatization of algebra, the definition of algebraic objects as pure abstractions, in the language of set theory. This axiomatic approach, when it reached full maturity a half-century later, became the foundation of the modern algebraic point of view.

The notion of an ideal is not an easy one to communicate to nonmathematicians because illuminating examples do not come easily to hand. An ideal is, first of all, a subring of a ring, a ring within a ring. It is, therefore, a family of numbers (or polynomials or whatever other objects the parent ring is composed of), closed under addition, subtraction, and multiplication, imbedded in a larger family of the same type.

An ideal is not just any old subring, though. It has this peculiarity: If you take any one of its elements and multiply this element by one from the larger ring, the result is bound to be within the subring.

Taking as the most familiar ring, an example of an ideal in would be: All multiples of some given number. Suppose we take the number 15, for example. Here is an ideal:

The ideal consists of all integers of the form 15m, where m is any integer whatsoever. Plainly the ideal is closed under addition, subtraction, and multiplication. And, as advertised, if you multiply any element from the ideal, say 30, by any element from the larger ring , say 2, the answer is in the ideal: 60.

It would be very nice if I could expand on this by saying: Now take any two numbers from and form all linear combinations of them. Take the numbers 15 and 22, for instance, and form all possible numbers 15m + 22n, where m and n are any integers whatsoever. That would be a more interesting ideal.

Unfortunately, nothing comes of this when working with because is just too simple in its structure. If you let m and n roam freely over , 15m + 22n takes every possible integer value, as can easily be proved.¹²² So the “ideal” you get is just the whole of . If, instead of 15 and 22, I had chosen two numbers with a common factor, say 15 and 21, I should just have gotten the ideal generated by 3, their greatest common divisor. So the kind of ideal shown above is the only kind in , other than itself (and the trivial ideal consisting of just zero). Ideals in are not, in fact, very interesting.

There is a way to say that in formal algebraic language. In any ring the set of multiples of some particular element a is called the principal ideal generated by a. A ring like , in which every ideal is a principal ideal, is called a principal ideal ring. In a ring that is not a principal ideal ring, you can indeed generate ideals by picking two or more elements a, b, … and running all possible combinations of them; am + bn + …. That would be called “the ideal generated by a, b, …” One way to classify rings, in fact, is by examining the way a ring’s ideals are generated. There is, for example, an important type of ring called a Noetherian ring, all of whose ideals are generated by a finite number of elements each.

In rings of complex numbers, ideals become very interesting indeed. Dedekind gave the abstract definition of an ideal—the one I just gave—and then applied it to a wide class of complex-number rings, a much wider class than Kummer had worked with. By doing so he was able to create definitions of “prime,” “divisor,” “multiple,” and “factor” appropriate to any ring at all.

These definitions were expressed in a way more general than any mathematician had attempted before. Dedekind did not completely detach himself from the realm of numbers, but he introduced his mathematical objects—field, ring (he calls it an “order”), ideal, module (a vector space whose scalars are taken from a ring instead of a field)—with defining axioms, as modern algebra textbooks do. Because he did not have the vocabulary of modern set theory to work with, Dedekind’s definitions do not look very modern, but he was on the right track.

I shall have more to say about ideals in the next chapter, when I cover algebraic geometry.

§12.7 Once Dedekind’s approach had been broadcast and accepted and the concept of an ideal made familiar, it became clear that rings could have interesting internal structure, like groups. That was when ring theory took off. It was still not thought of as ring theory, though.

The people who used it always had some particular application in mind: geometric, analytic, number-theoretic, or most often algebraic—I mean, concerned with polynomials. It was not until the Lady of the Rings came along after World War I that a coherent theory emerged, embracing all these areas and setting them on a firm axiomatic footing.

I shall introduce that lady in the next section. In the 40 years that elapsed between Dedekind’s work and hers, the theory was of course pushed forward by numerous mathematicians, including some great ones. The most interesting aspects of those efforts, though, were geometric and so belong in my next chapter. Here I am only going to mention one name from ring theory during that period, for the intrinsic interest of the man and his life.

The name is Emanuel Lasker, and he is mainly remembered not for mathematics but for chess. He was in fact world chess champion for 27 years, 1894–1921—the longest anyone has held that title.

Lasker was born in 1868 in that region of eastern Germany that became part of western Poland after the border rearrangements that followed World War II. His family was Jewish, his father a cantor in the synagogue of their little town, then named Berlinchen, now Barlinek. Lasker learned chess from his older brother and by his early teens was making pocket money by playing chess in the town’s coffee houses. He rose fast in the world of chess, winning his first tournament in Berlin at age 20 and becoming world champion at 25 in a series of matches played in North America (New York, Philadelphia, Montreal) against the reigning champion William Steinitz.

Lasker’s mathematical education was thorough but was interrupted by his chess activities. After attending the universities of Berlin, Göttingen, and Heidelberg, he studied under David Hilbert at Erlangen University (Germany) from 1900 to 1902 and got his doctorate in that latter year at age 33. His main contribution to ring theory was the rather abstruse notion of a primary ideal, somewhat analogous to the powers of primes that you get when you factorize an integer (for example, 6776 = 2³ × 7 × 11²). There is a type of ring called

a Lasker ring, and a key theorem, the Lasker–Noether theorem, about the structure of Noetherian rings.

Lasker’s life ended sadly. He and his wife had settled down to a comfortable retirement in Germany when Hitler came to power in 1933. The Nazis confiscated all of the Laskers’ property and drove them penniless out of their homeland. Emanuel Lasker, in his mid-60s, had to take up tournament chess again. He lived in England for two years and then moved to Moscow. When Stalin’s Great Purge began swallowing up his Russian friends, he moved to New York, dying there in 1941.¹²³

§12.8 Noetherian rings, the Lasker–Noether theorem—obviously there is a person named Noether in this story somewhere. There are in fact two, a lesser and a greater. The lesser was the father, Max Noether. The greater was his daughter Emmy, who brought together all that had been done in the 40 years since Dedekind’s ground-breaking work and transformed it into modern ring theory.

Max Noether was a professor of mathematics in the south German town of Erlangen, just north of Nuremberg. Emmy was born there in 1882. Her career must be seen in the context of the German empire in which she grew up, the empire of Bismarck (prime minister and chancellor to 1890) and Wilhelm II (German emperor—Kaiser—from 1888 to 1918). Wilhelmine Germany was an exceptionally misogynist society, even by late 19th-century standards. The German expression Kinder, Kirche, Küche (children, church, kitchen), supposedly identifying a woman’s proper place in society, is I think known even to people who don’t speak German. It was used approvingly of the attitude displayed by Wilhelm II’s lumpish consort, the Empress Augusta Victoria, except that on her lips it was supposed to have been uttered as Kaiser, Kinder, Kirche, Küche. For further insights into this topic, I recommend Theodor Fontane’s 1895 novel Effi Briest. Every literate person is familiar with the great French and Russian portrayals of anguished, transgressing 19th-century womanhood,

Flaubert’s Madame Bovary (1856) and Tolstoy’s Anna Karenina (1877), but few know the German entry in this field, Fontane’s dry, quiet little masterpiece.¹²⁴

Thus, when Emmy Noether decided, around age 18, to take up pure mathematics as a career, she had set herself to climb a steep mountain. This was so even though she had the advantage of a mathematician father, a professor at a prestigious university. In 1900, when Noether made her decision, women were allowed to sit in on university classes only as auditors and only with the professor’s permission. Emmy Noether accordingly sat in on math classes at Erlangen, 1900–1902, then at Göttingen, 1903–1904.

By 1907, there had been some modest reforms, and Noether was awarded a doctorate by Erlangen, only the second doctorate in mathematics given to a woman by a German university. The “habilitation” degree, however, the second doctorate that would have allowed her to teach at university level, was still not open to women. For eight years she worked at Erlangen as an unpaid supervisor of doctoral students and occasional lecturer. There was nothing to stop her publishing, and she quickly became known for brilliant work in mathematics.

These were the years following Albert Einstein’s unveiling of his special theory of relativity in 1905. Einstein was absorbed in trying to work out his general theory, which aimed to bring gravitation under the scope of his arguments. There were, though, some difficult problems to be overcome. In June and July of 1915 Einstein presented his general theory, unresolved problems and all, in some lectures at Göttingen University. Einstein noted of this event: “To my great joy I succeeded in convincing Hilbert and Klein.”

This was an occasion for joy indeed. David Hilbert and Felix Klein were, even at this fairly late point in their respective careers (Hilbert was 53, Klein 66), two giants of mathematics, while Einstein—he was 36—was still not far beyond the wunderkind stage. Hilbert and Klein had, of course, followed the development of Einstein’s ideas with interest before he came to lecture in 1915. Now “convinced” (convinced,

presumably, that Einstein was on the right lines), they gave their attention to the outstanding problems in the general theory. They knew of some work Emmy Noether had done in the relevant areas and invited her to Göttingen.

(Those relevant areas concerned invariants in certain transformations, ideas I shall clarify in the following pages. The key transformation in relativity theory is the Lorentz transformation, which tells us how coordinates—three of space, one of time—change when we pass from one frame of reference to another. Invariant under this transformation is the “proper time,” x² + y² + z² − c²t², at least at the infinitesimal level required to make calculus work.)

Noether duly arrived at Göttingen, and within a matter of months she produced a brilliant paper resolving one of the knottier issues in general relativity and providing a theorem still cherished by physicists today. Einstein himself praised the paper. Emmy Noether had arrived.

§12.9 Emmy Noether was now known as a first-class mathematician, but her professional troubles were not yet over. World War I was into its second year—Emmy’s younger brother Fritz (another mathematician) was in the army. Göttingen, though liberal by the standards of Wilhelmine universities, still balked at putting a woman on the faculty. David Hilbert, a broad-minded man who judged mathematicians by nothing but their talent, fought valiantly for Noether but without success.

Some of the arguments on both sides have become legendary among mathematicians. The faculty: “What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?” Hilbert: “I do not see that the sex of a candidate is an argument against her admission as a Privatdozent [that is, a lecturer supported from fees paid to him by students]. After all, we are a university, not a bathing establishment.”¹²⁵

Hilbert’s solution to the Noether problem was characteristic: He announced lecture courses in his own name and then allowed Noether to give them.

In the general liberalization of German society following defeat in World War I, however, it at last became possible for a woman to “habilitate” and get a university teaching position, if only of the Privatdozent variety, dependent on students paying their lecture fees. Noether duly habilitated in 1919. In 1922, she actually got a salaried position at Göttingen, though she had no tenure, and the meager salary was soon obliterated by hyperinflation.

It was during these early postwar years that Noether gathered up all the work that had been done on rings and turned it into a coherent abstract theory. Her 1921 paper Idealtheorie in Ringbereichen (“Ideal Theory in Ring-Fields”—terminology was not yet settled) is considered a landmark in the history of modern algebra, not only laying out key results on the inner structure of commutative rings¹²⁶ but providing an approach to the topic that was quickly taken up by other algebraists, the strictly axiomatic approach that became “modern algebra.”

Van der Waerden: “At Göttingen I had above all made the acquaintance of Emmy Noether. She had completely redone algebra, much more generally than any study made until then….”

By the early 1930s, Emmy Noether was at the center of a vigorous group of researchers at Göttingen. She still held a low-level position, ill paid and without tenure, but her power as a mathematician was not in doubt. Noether did not at all conform to the standards of femininity current in that time and place, though—nor, it must be said in fairness to her colleagues, any other time and place. She was stocky and plain, with thick glasses and a deep, harsh voice. She wore shapeless clothes and cropped her hair. Her lecturing style was generally described as impenetrable. Her colleagues regarded her with awe and affection nonetheless, though since they were all male, and Kaiser

Wilhelm’s Germany was only a dozen or so years in the past, the affection expressed itself in ways that would not be accepted nowadays.

Hence all the disparaging quips, not meant unkindly at the time, that have become part of mathematical folklore. Best known is the reply by her colleague Edmund Landau, when asked if he did not agree that Noether was an instance of a great woman mathematician: “Emmy is certainly a great mathematician; but that she is a woman, I cannot swear.” Norbert Wiener described her somewhat more generously as “an energetic and very nearsighted washerwoman whose many students flocked around her like a clutch of ducklings around a kind, motherly hen.” Hermann Weyl expressed the common opinion most gently: “The graces did not preside at her cradle.” Weyl also tried to take the edge off the appellation Der Noether (der being the masculine form of the definite article in German): “If we at Göttingen … often referred to her as Der Noether, it was … done with a respectful recognition of her power as a creative thinker who seemed to have broken through the barrier of sex … She was a great mathematician, the greatest.”

Ill paid and untenured as her position at Göttingen was, Noether lost it when the Nazis came to power in the spring of 1933. Having been once barred from university teaching for being a woman, she was now more decisively barred for being a Jew. The appeals of her Gentile colleagues and ex-colleagues—led, of course, by Hilbert—counted for nothing.

During the Nazi period, there were two common avenues of escape for Jewish or anti-Nazi intellectual talents: to the USSR or to the United States. Emmy’s brother Fritz chose the former, taking a job at an institute in Siberia. Emmy went the other way, to a position at Bryn Mawr College in Pennsylvania. Her English was passable, she was only 51, and the college was glad to acquire such a major mathematical talent. Alas, after only two years Emmy Noether died of an embolism following surgery for removal of a uterine tumor. Albert

Einstein wrote her obituary for the New York Times,¹²⁷ from which the following: