In spite of social prejudices and discrimination against women and Jews, Emmy Noether became a renowned mathematician through her groundbreaking research and discoveries on abstract algebra, invariant theorems, ideal theory and the theory of noncommutative algebras. The Noether School, Noether Boys and Noetherian Rings reflect the importance and impact that her discoveries have in the field of mathematics and physics.
In an unusual departure from the traditional way of referring to mathematicians by their last name, Amelie Emmy Noether was warmly known as Emmy in the mathematical community.
Emmy was born on 23 March 1882 in the small university town of Erlangen in Bavaria. She had three younger brothers, the oldest named Fritz. Her mother, Ida Amalia Kaufmann, a talented pianist, was the daughter of a prosperous merchant. Her father, Max Noether, was a distinguished mathematician, a professor at the University of Erlangen who played an important role in the theory of algebraic functions, as well as in both Emmy’s and Fritz’s professional lives. Indeed, Emmy later expanded on her father’s interest in algebraic functions, and ultimately did some of her most important work in that subject. Emmy built on her father’s residue theorem: “during the 1920s she fitted this theorem into her general theory of ideals in arbitrary rings, helping to further establish the axiomatic and integrative tendencies of abstract algebras.” [e8]
Emmy’s work in abstract algebra was groundbreaking, providing significantly new directions in mathematics. It led her to discover the key condition for rings, the ascending-chain condition; such rings are now known as Noetherian rings. By 1918 she was recognized for the “extreme generality and abstractness of approach that [would eventually be] seen as her most distinguishing characteristic.” [e9]
Emmy’s home life had a strong influence on her. Hermann Weyl, an eminent mathematician, friend, and later a colleague of Emmy’s, remembered Max Noether as a “very intelligent, warm-hearted harmonious man of many-sided interests and a sterling education” [e14]. Weyl’s observation of Emmy was that “[she] herself was, if I might say so, warm like a loaf of bread. There irradiated from her a broad, comforting, vital warmth” [e14]. Their home was often filled with mathematicians and thought-provoking discussions, which sparked an interest in mathematics in Emmy. Her warm home life was reflected by Emmy when she began to teach — she loved her students; they would come to her apartment for tea, take long walks together, always talking mathematics [e11]. Her students became known as the “Noether Boys”.
As a child, Emmy received the education typical for girls of her time and income level, learning household chores, music and dancing, as well as reading, writing and arithmetic. She did not stand out academically, although she displayed at times an aptitude for logic. She did become proficient in French and English, enough to become certified to teach languages in Bavarian girls’ schools. But Emmy’s interest lay in mathematics, and she chose to continue her education at the University of Erlangen, where her brother Fritz was already a student. Women could attend lectures only unofficially and only with the express permission of the lecturer. Nevertheless, she passed her graduation examination in July 1903, and the following semester she went to the University of Göttingen. When the restrictions against female enrollment were dropped, Emmy transferred back to Erlangen.
A frequent visitor to the Noether home was Paul Gordan, a family friend and professor at Erlangen. Under Gordan’s supervision she wrote her dissertation [e1], explaining “new properties she discovered of ternary biquadratic forms, algebraic operators related to polynomials with three variables in which the exponents in every item add up to four” [e15]. (Emmy dismissed her thesis as a “jumble of formulas” [e8].) She successfully defended her thesis, and in December 1907, at the age of 26, received her doctorate in mathematics summa cum laude. She was one of the first German women to earn a Ph.D. in any subject [e15].
Having overcome restrictions on women’s education, Emmy was now faced with the restrictions placed on women’s employment, particularly in the academic sphere. The natural next step for a recipient of a doctoral degree would have been the Habilitation (an independent thesis at a level above the Ph.D.), the final step necessary for a candidate to become qualified to lecture at German universities. As a woman, Emmy was barred from this path. Instead, she taught at the Mathematical Institute of Erlangen for several years without pay or a title, sometimes substituting for her father when he was too ill to teach. She also began to conduct her own research; although an unofficial faculty member, two students received their doctorates under her direction [e15]. Gordan’s successor, algebraist Ernst Fischer, was particularly influential in transitioning Emmy’s focus from Gordan’s formalist approach to her now-famous axiomatic approach. Weyl’s observation is interesting [e14]:
It is queer enough that a formalist like Gordan was the mathematician from whom her mathematical orbit set out; a greater contrast is hardly imaginable between her first paper, the dissertation, and her works of maturity; for the former is an extreme example of formal computations and the latter constitute an extreme and grandiose example of conceptual axiomatic thinking in mathematics.
Two papers she wrote in 1913, “Rational function fields” [e2], and an extended paper on “Fields and systems of rational functions” [e3], along with her doctoral dissertation, “established Emmy’s reputation in the field of invariant theory, which is where researchers study properties that remain fixed when an object is subjected to modifying transformations” [e15].
In 1915, Hilbert and Felix Klein invited Emmy to return to the University of Göttingen to work with them on the research they were conducting on applications of invariant theory. Einstein, a physics professor at the University of Berlin, had already formulated his theory on relativity, and Klein and Hilbert were “trying to determine the field equations for general relativity to describe the properties of a gravitational field surrounding a given mass” [e15].
During the next four years, Emmy wrote nine papers on various aspects of invariant theory. One on the Galois group of polynomial equations “represented the most significant contribution to the solution of this classic problem” [e15].
Emmy’s 1918 paper on “Invariant variational problems” [e4] is her most widely known work. It contains Noether’s Theorem, which explains the relationship between physical symmetry and conservation laws, and is considered to be of great importance in the development of modern physics.
It shows how symmetries in a physical system lead to conserved quantities. In the paper, Emmy considers physical systems based on an action principle. That is, the laws of motion come from minimizing some action functional. (This is the case for most mechanical systems.) She then adds the assumption that there is a continuous group of symmetries of the physical system, i.e., the action of a Lie group on the configuration space, leaving the action functional unchanged. From this, she derives explicit quantities that are constant along the physical trajectories. The number of such independent quantities equals the dimension of the Lie group.
An example is a mechanical system of \( n \) particles whose interactions are unchanged by a simultaneous translation in 3-space of all of the particles, or by the rotation around a point of all of the particles. The conserved quantities coming from the translation symmetries are the total momenta. The conserved quantities coming from the rotations are the total angular momenta.
Emmy’s paper is framed in the abstract setting of variational calculus. She does not discuss physical examples, although her work was inspired by the ongoing efforts of Hilbert to derive the laws of general relativity from an action principle.
However, due to the objections of other faculty members, Emmy’s role was again restricted. The professors felt that it would be humiliating to require male students to be taught by a female professor. She was still not paid for her work, nor did she have an official title. She gave lectures that were advertised under Hilbert’s name (“with the assistance of Dr. E. Noether”). Hilbert was so frustrated and angry over being unable to secure a paid position for Emmy at Göttingen that he declared at a faculty meeting, “I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, we are a university and not a bathing establishment” [e14].
In 1919, after the end of the war, conditions changed and Emmy was given a title of “Professor”; she could now lecture under her own name, but with no salary. In 1923, thanks to her close mathematical colleagues, she received an official commission to teach, that would pay her a small salary. It was during this time that the “Noether Boys” became a phenomenon. This was a group of ten male students who admired her and her mathematical brilliance, and who received their doctoral degrees in mathematics under her supervision.
In 1920 she co-authored (with Schmeidler) a paper on differential operators [e5] that changed the face of algebra. Its immediate influence was minor, but looking back this is where her conceptual axiomatic way of thinking truly surfaces.
Emmy’s papers from 1920 to 1926 emphasized abstract properties of groups, rings, fields, ideals, and modules, rather than the specific objects themselves. Her monumental 1921 paper “The theory of ideals in ring domains” [e6] has become the basis for commutative ring theory. She introduced what are now commonly referred to as Noetherian rings and Noetherian ideals.
Emmy’s research on abstract algebra was a magnet for many students and professors. She and her group became known as the Noether School, and as a result the Mathematical Institute at Göttingen became the world’s most respected and influential center for mathematical research.
Over a period of time, beginning with the winter of 1925–26 when she visited Holland and talked with L. E. J. Brouwer and P. S. Aleksandrov, and later in courses and talks with Heinz Hopf, Emmy influenced topologists to study homology groups as quotients of the group of cycles by the subgroup of boundaries, rather than simply computing Betti numbers [e12]. Aleksandrov and Hopf, in the preface to their 1935 book Topologie, I [e7], wrote:
Emmy Noether’s overall mathematical insight was not limited to algebra, her particular specialty, but exerted an enlivening influence on anyone who had mathematical contact with her. […] [W]e follow the trend towards a thorough “algebraicization” of topology, based on group theory, which goes directly back to Emmy Noether. Today, this development appears self-evident; eight years ago it was not. It took Emmy Noether’s energy and temperament to make algebraic thinking part of the topologist’s repertoire, and to allow algebraic problems and methods to play the role they do in topology today.
During 1927–1935, her abstract focus was redirected to the study of noncommutative algebras. During this period, she wrote 13 papers in this area, one of which Hermann Weyl regarded as “a high watermark in the development of algebra” [e15].
In 1932, two events affirmed the respect for Emmy within the mathematics community: she was awarded the Alfred Ackermann-Taubner Memorial Award for the advancement of mathematical sciences, and in September was invited to deliver one of the main addresses at the International Mathematical Congress (ICM) in Switzerland.
When the Nazis came to power in Germany, Emmy was again barred from teaching at the university. She had three strikes against her: she was Jewish, an intellectual woman, and a philosophical activist. “The serious struggles that shook Germany during these years helped shape Emmy’s philosophy as a pacifist, an attitude she held very strongly for the rest of her life” [e8]. In late 1933 and with a grant from the Rockefeller Foundation, Emmy went to Bryn Mawr College as a professor. She also began lecturing at the Institute for Advanced Study in Princeton. (Her brother Fritz went to the Research Institute for Mathematics and Mechanics in Tomsk, Siberia.)
In early 1935, however, Emmy had to have an operation to remove a tumor in her pelvis. Doctors during the operation found a large ovarian cyst, which they removed, and two smaller tumors appeared to be benign. After four days of apparently normal post-operative recuperation, Emmy died unexpectedly. She was only 53 years old and at the peak of her productive power and technical skill.
Her old friend who dearly loved her, Hermann Weyl, delivered the memorial address at Bryn Mawr College on April 26, 1935. Here’s an excerpt:
It was only too easy for those who met her for the first time, or had no feeling for her creative power, to consider her queer and to make fun at her expense. She was heavy of build and loud of voice, and it was often not easy for one to get the floor in competition with her. She preached mightily, and not as the scribes. She was a rough and simple soul, but her heart was in the right place. Her frankness was never offensive in the least degree. In everyday life she was most unassuming and utterly unselfish; she had a kind and friendly nature. Nevertheless she enjoyed the recognition paid her; she could answer with a bashful smile like a young girl to whom one had whispered a compliment. No one could contend that the Graces had stood by her cradle; but if we in Göttingen often chaffingly referred to her as “der Noether” (with the masculine article), it was also done with a respectful recognition of her power as a creative thinker who seemed to have broken through the barrier of sex. She possessed a rare humor and sense of sociability; a tea in her apartments could be most pleasurable. But she was a one-sided who was thrown out of balance by the overweight of her mathematical talent. […] The memory of her work in science and of her personality among her fellows will not soon pass away. She was a great mathematician, the greatest, I firmly believe, that her sex has ever produced and a great woman.
A more complete abstract can be read on this site.
Her friend and colleague, Albert Einstein, said
In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.
The complete letter follows.
Einstein’s letter to the New York Times
published 3 May 1935
To the Editor of The New York Times:
The efforts of most human beings are consumed in the struggle for their daily bread, but most of those who are, either through fortune or some special gift, relieved of this struggle are largely absorbed in further improving their worldly lot. Beneath the effort directed toward the accumulation of worldly goods lies all to frequently the illusion that this is the most substantial and desirable end to be achieved; but there is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual’s own feeling, thinking and acting. The genuine artists, investigators and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, none the less the fruits of their endeavors are the most valuable contributions which one generation can make to its successors.
Within the past few days a distinguished mathematician, Professor Emmy Noether, formerly connected with the University of Göttingen and for the past two years at Bryn Mawr College, died in her fifty-third year. In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulae are discovered necessary for the deeper penetration into the laws of nature.
Born in a Jewish family distinguished for the love of learning, Emmy Noether, who, in spite of the efforts of the great Göttingen, Hilbert, never reached the academic standing due her in her own country, none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators. Her unselfish, significant work over a period of many years was rewarded by new rulers of Germany with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies. Farsighted friends of science in this country were fortunately able to make such arrangements at Bryn Mawr College and at Princeton that she found in America up to the day of her death not only colleagues who esteemed her friendship but grateful pupils whose enthusiasm made her last years the happiest and perhaps the most fruitful of her entire career.
Albert Einstein,
Princeton University, May 1, 1935.
Seventy-six years after the untimely death of Emmy Noether, it is difficult to say much new, given the wealth of material written about her. The most complete work is of course her collected works [e10], with its fine introduction by Nathan Jacobson. Also, the book [e9] edited by Brewer and Smith has many interesting articles written on the occasion of her one-hundredth birthday. A 1996 conference proceedings from Bar-Ilan University [e13] has fine articles on Emmy’s work and influence. For those interested in Emmy’s influence on topology, [e16] is a good place to start.
