# Emmy Noether

# Emmy Noether

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

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. □

# Noether, Emmy

# NOETHER, EMMY

Emmy Noether was a very important mathematician whose work profoundly influenced twentieth-century physics. Her 1918 paper "Invariante Variationsprobleme" contains theorems and their converses that reveal deep, fundamental connections between symmetries and conservation laws. Although she knew that these theorems were of great importance for physics, she regarded the work as something of a departure from the main line of her research. The work was done just after the completion of the general theory of relativity, when Albert Einstein, her Göttingen colleague David Hilbert, and others were seeking a principle of conservation of energy in the general theory of relativity. Her work clarified the issue of conservation of energy and solved the problem of the apparent absence of the law in Einstein's theory. After completing this work, she returned to her main line of research, which was development of abstract algebra. In the publication of her *Collected Works,* editor Nathan Jacobson writes, "Abstract algebra is one of the most distinctive innovations of twentieth century mathematics, and it is largely due to her" (Jacobson, 1983). Concepts and methods of modern algebra are now widely used in all areas of physics. Noether is distinguished as a contributor to physics not only for her work on symmetries and conservation laws but also for her contributions to modern abstract algebra whose importance for twentieth-century physics cannot be overstated.

Emmy Noether was registered as a student in the University of Erlangen in 1904, the first year women were admitted. She is an outstanding example of the fact that, after women had been excluded for centuries, female scientific genius emerged as soon as women were allowed to study in institutions of higher learning. Noether was born March 23, 1882, in Erlangen, Germany. Her father, Max Noether, was Professor of Mathematics at the University of Erlangen. His father was the first in the family to take up academic studies, but he became a merchant, and the family assumed that Max's mathematical talent came from his mother's side. Max Noether was a distinguished and highly respected mathematician who became even more distinguished in later years as the father of Emmy. Emmy attended the Municipal School for Higher Education of Daughters in Erlangen from age seven to age fifteen, when her formal schooling ended, as was normal for girls at that time. She then studied French and English and was certified to teach in girls' schools. To further her education, she sought permission to audit lectures at the University of Erlangen. This was granted, but at the discretion of the lecturer; some professors refused to lecture when a woman was present. In the winter of 1903–1904, she went to Göttingen to attend the lectures of the great mathematicians there. She returned to Erlangen in 1904 when, after years of debate, the university finally admitted women. She wrote a doctoral thesis under the direction of Paul Gordan and was awarded Ph.D. summa cum laude in 1908.

For more than a decade after receiving her Ph.D., Emmy Noether worked unpaid at the University of Erlangen, and later at Göttingen, teaching and doing mathematical research. During this period, she published fifteen papers in important mathematical journals, became a member of the prestigious Cicolo Mathematico di Palermo and the Deutsche Mathematiker Vereinigung (German Association of Mathematics [DMV]), and gave two lectures to the DMV. David Hilbert and Felix Klein invited

her to join their group at the University of Göttingen, then a world center for mathematics and physics. In 1915, she went to Göttingen after her father died. Prior to his death, his health had not been good, and she had been filling in for him as lecturer at the University of Erlangen. She was refused appointment as lecturer (Privatdocent) by the University of Göttingen in spite of the very strong recommendations of mathematicians such as Hilbert and Klein who were two of the university's most distinguished scholars. The reason the university refused to appoint her was because she was a woman. This so enraged Hilbert that he stormed out of a faculty meeting saying, "I do not see that the sex of a candidate is an argument against her admission as *Privatdocent.* After all, we are a university, not a bathing establishment."

On July 16, 1918, Noether's paper "Invariante Variationsprobleme" was read to the Königliche Gesellschaft der Wissenschaften zu Göttingen (Royal Society of Sciences of Göttingen) by Felix Klein. Presumably Klein presented it because Noether was not a member of the Society; it seems likely she wasn't even there when the paper was read. Records of the Society were lost in World War II, and it is not known when women were first admitted; counterpart societies in London and Paris did not admit women until after World War II. For example, the Royal Society (London) elected its first female member in 1945, and the Académie des Sciences of Paris did so in 1962.

This paper was immediately seen to be of enormous importance because it shows with great generality logical connections between symmetries and dynamical properties of the fundamental forces of nature. The results led to a deeper understanding of the principle of conservation of energy and of a vast variety of other conservation laws as well. Since the paper gives proofs of theorems and their converses, the insight it provides has led to discoveries of new symmetries of nature following empirical discoveries of new conservation laws. Examples include gauge field symmetries in the Standard Model of particle physics. In 1919 Noether was given *Habilitation* and was finally able to lecture as Privatdocent and be paid. In 1922 she became *Lehrauftrag für Algebra und nicht-beamteter ausserordentlicher Professor* and was paid a salary, although this was not an ordinary faculty appointment.

Historians of mathematics date the creation of modern abstract algebra to the work of Emmy Noether and Emil Artin and their school during 1921 to1933. Prominent mathematicians came from all over Germany and abroad to consult with Noether and attend her lectures. It was in 1921 that she published "Ideal Theorie in Ringbereichen," which is regarded as a truly monumental work, the first paper published on this vast subject. A pillar of this subject is her 1929 paper "Hyperkomplexe Grössen und Darstellungstheorie," which gave a general representation theory of groups and algebras, valid for arbitrary ground fields.

In 1933, when Adolf Hitler came to power, all Jews were dismissed from the University of Göttingen, and she could no longer teach there. A world-renowned mathematician, she had only two job offers. They were from two leading women's colleges, Bryn Mawr in Pennsylvania and Somerville College in Oxford, UK. Lacking financial resources, they could not offer her a secure position. Somerville offered her room and board and a small stipend. The Bryn Mawr job was subsidized in part by the Rockefeller Foundation. She went to Pennsylvania and was also invited to lecture at the Institute for Advanced Study in Princeton, New Jersey. Weekly, she took the train from Bryn Mawr to Princeton to lecture there. As in Göttingen, her lectures were very well attended. Presumably, if she had not been female, she would have been invited to be a member of the institute.

Following what was expected to be a minor surgery, Emmy Noether died at Bryn Mawr in April1935. In memoriam Albert Einstein wrote, "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 . . . Pure mathematics is, in its way, the poetry of logical ideas. . . . In this effort toward logical beauty, spiritual formulas are discovered necessary for deeper understanding of laws of nature" (Einstein, 1935). Einstein may have been referring, in part, to formulas Noether gave in her 1918 paper. They do indeed yield deeper understanding of laws of nature. What must have been most impressive to Einstein shortly after he completed the general theory of relativity was the deeper understanding of the law of conservation of energy her work offered. Conservation of energy is one of the most important conservation laws in physics. In the early days of the general theory, Einstein, Hilbert, and others were perplexed by the apparent absence of a law of conservation of energy in Einstein's theory. Noether's theories solved this problem. Her Theorem I shows that space-time symmetry implies local energy conservation in classical field theories that are nonrelativistic or governed by the special theory of relativity—a familiar result. On the other hand, in the presence of gravitational fields, the general theory applies, and there is no such local energy conservation law. Instead, her Theorem II shows that space-time symmetry implies a different formula. This is necessary to accommodate gravitational radiation and the principle of equivalence. These results give deeper meaning to the principle of conservation of energy. What is truly extraordinary about her theorems is that they are very general and apply to a vast variety of symmetries and conservation laws.

*See also:*Conservation Laws; Symmetry Principles

## Bibliography

Bourbaki, N. *Elements of Historical Mathematics,* translated by J. Meldrum (Springer-Verlag, Berlin, 1994).

Byers, N. "The Life and Times of Emmy Noether: Contributions of Emmy Noether to Particle Physics" in *History of Original Idea and Basic Discoveries in Particle Physics, II,* edited by B. Newman and T. Ypsilantis (Plenum, New York, 1996).

Byers, N. "E. Noether's Discovery of the Deep Connection between Symmetries and Conservation Laws." *Israel Mathematical Conference Proceedings***12** , 215–230 (1999).

Dick, A. *Emmy Noether (1882–1935),* translated by H. I. Blocher (Birkhauser, Boston, 1981).

Einstein, A. "Letter to the Editor." *New York Times* (May 5, 1935).

Jacobson, N. Introduction to *Emmy Noether, Collected Papers,* edited by N. Jacobson (Springer-Verlag, Berlin, 1983).

UCLA. "Contributions of 20th Century Women to Physics." <http://www.physics.ucla.edu/~ewp/>.

*Nina Byers*

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