**Emmy Noether (1882-1935) was a world-renowned mathematician whose innovative approach to modern abstract algebra inspired colleagues and students who emulated her technique.**

Dismissed from her university position at the beginning of the Nazi era in Germany—for she was both Jewish and female—Noether emigrated to the United States, where she taught in several universities and colleges. When she died, Albert Einstein eulogized her in a letter to New York Times as "the most significant creative mathematical genius thus far produced since the higher education of women began."

Noether was born on March 23, 1882, in the small university town of Erlangen in southern Germany. Her first name was Amalie, but she was known by her middle name of Emmy. Her mother, Ida Amalia Kaufmann Noether, came from a wealthy family in Cologne. Her father, Max Noether, a professor at the University of Erlangen, was an accomplished mathematician who worked on the theory of algebraic functions. Two of her three younger brothers became scientists—Fritz was a mathematician and Alfred earned a doctorate in chemistry.

Noether's childhood was unexceptional, going to school, learning domestic skills, and taking piano lessons. Since girls were not eligible to enroll in the gymnasium (college preparatory school), she attended the Städtischen Höheren Töchterschule, where she studied arithmetic and languages. In 1900 she passed the Bavarian state examinations with evaluations of "very good" in French and English (she received only a "satisfactory" evaluation in practical classroom conduct); this certified her to teach foreign languages at female educational institutions.

Instead of looking for a language teaching position, Noether decided to undertake university studies. However, since she had not graduated from a gymnasium, she first had to pass an entrance examination for which she obtained permission from her instructors. She audited courses at the University of Erlangen from 1900 to 1902. In 1903 she passed the matriculation exam, and entered the University of Göttingen for a semester, where she encountered such notable mathematicians as Hermann Minkowski, Felix Klein, and David Hilbert. She enrolled at the University of Erlangen where women were accepted in 1904. At Erlangen, Noether studied with Paul Gordan, a mathematics professor who was also a family friend. She completed her dissertation entitled "On Complete Systems of Invariants for Ternary Biquadratic Forms," receiving her Ph.D., summa cum laude, on July 2, 1908.

Noether worked without pay at the Mathematical Institute of Erlangen from 1908 until 1915, where her university duties included research, serving as a dissertation adviser for two students, and occasionally delivering lectures for her ailing father. In addition, Noether began to work with Ernst Otto Fischer, an algebraist who directed her toward the broader theoretical style characteristic of Hilbert. Noether not only published her thesis on ternary biquadratics, but she was also elected to membership in the Circolo Matematico di Palermo in 1908. The following year, Noether was invited to join the German Mathematical Society (Deutsche Mathematiker Vereinigung); she addressed the Society's 1909 meeting in Salzburg and its 1913 meeting in Vienna.

In 1915, Klein and Hilbert invited Noether to join them at the Mathematical Institute in Göttingen. They were working on the mathematics of the newly announced general theory of relativity, and they believed Noether's expertise would be helpful. Albert Einstein later wrote an article for the 1955 Grolier Encyclopedia, characterizing the theory of relativity by the basic question, "how must the laws of nature be constituted so that they are valid in the same form relative to arbitrary systems of co-ordinates (postulate of the invariance of the laws of nature relative to an arbitrary transformation of space and time)?" It was precisely this type of invariance under transformation on which Noether focused her mathematical research.

In 1918, Noether proved two theorems that formed a cornerstone for general relativity. These theorems validated certain relationships suspected by physicists of the time. One, now known as Noether's Theorem, established the equivalence between an invariance property and a conservation law. The other involved the relationship between an invariance and the existence of certain integrals of the equations of motion. The eminent German mathematician Hermann Weyl described Noether's contribution in the July 1935 Scripta Mathematica following her death: "For two of the most significant sides of the general theory of relativity theory she gave at that time the genuine and universal mathematical formulation."

While Noether was proving these profound and useful results, she was working without pay at Göttingen University, where women were not admitted to the faculty. Hilbert, in particular, tried to obtain a position for her but could not persuade the historians and philosophers on the faculty to vote in a woman's favor. He was able to arrange for her to teach, however, by announcing a class in mathematical physics under his name and letting her lecture in his place. By 1919, regulations were eased somewhat, and she was designated a Privatdozent (a licensed lecturer who could receive fees from students but not from the university). In 1922, Noether was given the unofficial title of associate professor, and was hired as an adjunct teacher and paid a modest salary without fringe benefits or tenure.

Noether's enthusiasm for mathematics made her an effective teacher, often conducting classroom discussions in which she and her students would jointly explore some topic. In Emmy Noether at Byrn Mawr, Noether's only doctoral student at Bryn Mawr, Ruth McKee, recalls, "Miss Noether urged us on, challenging us to get our nails dirty, to really dig into the underlying relationships, to consider the problems from all possible angles."

Brilliant mathematicians often make their greatest contributions early in their careers; Noether was one of the notable exceptions to that rule. She began producing her most powerful and creative work about the age of 40. Her change in style started with a 1920 paper on non-commutative fields (systems in which an operation such as multiplication yields a different answer foraxb than for b x a). During the years that followed, she developed a very abstract and generalized approach to the axiomatic development of algebra. As Weyl attested, "she originated above all a new and epoch-making style of thinking in algebra."

Noether's 1921 paper on the theory of ideals in rings is considered to contain her most important results. It extended the work of Dedekind on solutions of polynomials— algebraic expressions consisting of a constant multiplied by variables raised to a positive power—and laid the foundations for modern abstract algebra. Rather than working with specific operations on sets of numbers, this branch of mathematics looks at general properties of operations. Because of its generality, abstract algebra represents a unifying thread connecting such theoretical fields as logic and number theory with applied mathematics useful in chemistry and physics.

During the winter of 1928-29, Noether was a visiting professor at the University of Moscow and the Communist Academy, and in the summer of 1930, she taught at the University of Frankfurt. Recognized for her continuing contributions in the science of mathematics, the International Mathematical Congress of 1928 chose her to be its principal speaker at one of its section meetings in Bologna. In 1932 she was chosen to address the Congress's general session in Zurich.

Noether was a part of the mathematics faculty of Göttingen University in the 1920s when its reputation for mathematical research and teaching was considered the best in the world. Still, even with the help of the esteemed mathematician Hermann Weyl, Noether was unable to secure a proper teaching position there, which was equivalent to her male counterparts. Weyl once commented: "I was ashamed to occupy such a preferred position beside her whom I knew to be my superior as a mathematician in many respects." Nevertheless, in 1932, on Noether's fiftieth birthday, the university's algebraists held a celebration, and her colleague Helmut Hasse dedicated a paper in her honor, which validated one of her ideas on noncommutative algebra. In that same year, she again was honored by those outside her own university, when she was named cowinner of the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge.

The successful and congenial environment of the University of Göttingen ended in 1933, with the advent of the Nazis in Germany. Within months, anti-Semitic policies spread through the country. On April 7, 1933, Noether was formally notified that she could no longer teach at the university. She was a dedicated pacifist, and Weyl later recalled, "her courage, her frankness, her unconcern about her own fate, her conciliatory spirit were, in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace."

For a while, Noether continued to meet informally with students and colleagues, inviting groups to her apartment. But by summer, the Emergency Committee to Aid Displaced German Scholars was entering into an agreement with Bryn Mawr, a women's college in Pennsylvania, which offered Noether a professorship. Her first year's salary was funded by the Emergency Committee and the Rockefeller Foundation.

In the fall of 1933, Noether was supervising four graduate students at Bryn Mawr. Starting in February 1934, she also delivered weekly lectures at the Institute for Advanced Study at Princeton. She bore no malice toward Germany, and maintained friendly ties with her former colleagues. With her characteristic curiosity and good nature, she settled into her new home in America, acquiring enough English to adequately converse and teach, although she occasionally lapsed into German when concentrating on technical material.

During the summer of 1934, Noether visited Göttingen to arrange shipment of her possessions to the United States. When she returned to Bryn Mawr in the early fall, she had received a two-year renewal on her teaching grant. In the spring of 1935, Noether underwent surgery to remove a uterine tumor. The operation was a success, but four days later, she suddenly developed a very high fever and lost consciousness. She died on April 14th, apparently from a post-operative infection. Her ashes were buried near the library on the Bryn Mawr campus.

Over the course of her career, Noether supervised a dozen graduate students, wrote forty-five technical publications, and inspired countless other research results through her habit of suggesting topics of investigation to students and colleagues. After World War II, the University of Erlangen attempted to show her the honor she had deserved during her lifetime. A conference in 1958 commemorated the fiftieth anniversary of her doctorate; in 1982 the university dedicated a memorial plaque to her in its Mathematics Institute. During the same year, the 100th anniversary year of Noether's birth, the Emmy Noether Gymnasium, a coeducational school emphasizing mathematics, the natural sciences, and modern languages, opened in Erlangen.

## Further Reading on Emmy Noether

Brewer, James W., Emmy Noether: A Tribute to Her Life and Work, edited by Martha K. Smith, Marcel Dekker, 1981.

Kramer, Edna E., The Nature and Growth of Modern Mathematics, Princeton University, 1981, pp. 656-672.

Magill, Frank N., editor, Great Events from History II, Books International, 1991, pp. 650-654, 716-719.

Osen, Lynn M., Women in Mathematics, Massachusetts Institute of Technology, 1979, pp. 141-152.

Perl, Teri, Math Equals: Biographies of Women Mathematicians, Addison-Wesley, 1978, pp. 172-178.

Srinivasan, Bhama and Judith D. Sally, Emmy Noether in Bryn Mawr: Proceedings of a Symposium, Springer-Verlag, 1983.

Kimberling, Clark H., "Emmy Noether," in The American Mathematical Monthly, February, 1972, pp. 136-149.