Emmy Noether

Template:Short description Template:Featured article Template:Use dmy dates Template:CS1 config Template:Infobox scientist Amalie Emmy NoetherTemplate:Refn (Template:IPAc-en, Template:IPAc-en; {{#invoke:IPA|main}}; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental in mathematical physics.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Noether was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics.<ref name="einstein">Template:Cite news Transcribed online at the MacTutor History of Mathematics Archive.</ref>Template:Sfn Template:Sfn As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.<ref name="neeman_1999">Template:Citation in Template:Harvnb.</ref>

Noether was born to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen–Nuremberg, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years.Template:Sfn At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent.Template:Sfn

Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called the "Noether Boys". In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas; her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra. By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world. The following year, Germany's Nazi government dismissed Jews from university positions, and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania. There, she taught graduate and post-doctoral women including Marie Johanna Weiss and Olga Taussky-Todd. At the same time, she lectured and performed research at the Institute for Advanced Study in Princeton, New Jersey.Template:Sfn

Noether's mathematical work has been divided into three "epochs".<ref name=Weyl>Template:Harvnb</ref> In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields. Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics".Template:Sfn In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra".<ref name="weyl_128"/> In her classic 1921 paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains), Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications. She made elegant use of the ascending chain condition, and objects satisfying it are named Noetherian in her honor. In the third epoch (1927–1935), she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology.

BiographyEdit

Early lifeEdit

File:Erlangen 1916.jpg

Noether grew up in the Bavarian city of Erlangen, depicted here in a 1916 postcard.

Amalie Emmy Noether was born on 23 March 1882 in Erlangen, Bavaria.Template:Sfn She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from wealthy Jewish merchant families.Template:Sfn Her first name was "Amalie", but she began using her middle name at a young age and invariably continued to do so in her adult life and her publications.Template:Refn

In her youth, Noether did not stand out academically, although she was known for being clever and friendly. She was near-sighted and talked with a minor lisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at an early age.Template:Sfn She was taught to cook and clean, as were most girls of the time, and took piano lessons. She pursued none of these activities with passion, although she loved to dance.Template:Sfn

File:NoetherFamily MFO3120.jpg

Emmy Noether with her brothers Alfred, Fritz, and Robert, before 1918

Noether had three younger brothers. The eldest, Alfred Noether, was born in 1883 and was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later.Template:Sfn Fritz Noether was born in 1884, studied in Munich and made contributions to applied mathematics.Template:Sfn He was likely executed in the Soviet Union in 1941.<ref>Template:MacTutor.</ref> The youngest, Gustav Robert Noether, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928.Template:Sfn Template:Sfn

EducationEdit

Noether showed early proficiency in French and English. In the spring of 1900, she took the examination for teachers of these languages and received an overall score of sehr gut (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen–Nuremberg,Template:Sfn at which her father was a professor.Template:Sfn

This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing mixed-sex education would "overthrow all academic order".Template:Sfn One of just two women in a university of 986 students, Noether was allowed only to audit classes rather than participate fully, and she required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903, she passed the graduation exam at a Realgymnasium in Nuremberg.Template:Sfn Template:Sfn Template:Sfn

During the 1903–1904 winter semester, she studied at the University of Göttingen, attending lectures given by astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert.Template:Sfn

File:Paul Albert Gordan.jpg

Paul Gordan supervised Noether's doctoral dissertation on invariants of biquadratic forms.

In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded.Template:Sfn Noether returned to Erlangen and officially reentered the university in October 1904, declaring her intention to focus solely on mathematics. She was one of six women in her year (two auditors) and the only woman in her chosen school.Template:Sfn Under the supervision of Paul Gordan, she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms),Template:Sfn in 1907, graduating summa cum laude later that year.Template:Sfn Gordan was a member of the "computational" school of invariant researchers, and Noether's thesis ended with a list of over 300 explicitly worked-out invariants. This approach to invariants was later superseded by the more abstract and general approach pioneered by Hilbert.Template:Sfn Template:Sfn Although it had been well received, Noether later described her thesis and some subsequent similar papers she produced as "crap". All of her later work was in a completely different field.Template:Sfn Template:Sfn

University of Erlangen–NurembergEdit

From 1908 to 1915, Noether taught at Erlangen's Mathematical Institute without pay, occasionally substituting for her father, Max Noether, when he was too ill to lecture.Template:Sfn She joined the Circolo Matematico di Palermo in 1908 and the Deutsche Mathematiker-Vereinigung in 1909.Template:Sfn In 1910 and 1911, she published an extension of her thesis work from three variables to n variables.Template:Sfn

File:Emmy noether postcard 1915.jpg

Noether sometimes used postcards to discuss abstract algebra with her colleague, Ernst Fischer. This card is postmarked 10 April 1915.

Gordan retired in 1910,Template:Sfn and Noether taught under his successors, Erhard Schmidt and Ernst Fischer, who took over from the former in 1911.Template:Sfn According to her colleague Hermann Weyl and her biographer Auguste Dick, Fischer was an important influence on Noether, in particular by introducing her to the work of David Hilbert.Template:Sfn Template:Sfn Noether and Fischer shared lively enjoyment of mathematics and would often discuss lectures long after they were over; Noether is known to have sent postcards to Fischer continuing her train of mathematical thoughts.Template:Sfn Template:Sfn

From 1913 to 1916, Noether published several papers extending and applying Hilbert's methods to mathematical objects such as fields of rational functions and the invariants of finite groups.Template:Sfn This phase marked Noether's first exposure to abstract algebra, the field to which she would make groundbreaking contributions.Template:Sfn

In Erlangen, Noether advised two doctoral students:<ref name="MacTutorStudents">Template:MacTutor</ref> Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916.Template:Sfn Template:Sfn Despite Noether's significant role, they were both officially under the supervision of her father. Following the completion of his doctorate, Falckenberg spent time in Braunschweig and Königsberg before becoming a professor at the University of Giessen Template:Sfn while Seidelmann became a professor in Munich.<ref name="MacTutorStudents"/>

University of GöttingenEdit

Habilitation and Noether's theoremEdit

In the spring of 1915, Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein. Their effort to recruit her was initially blocked by the philologists and historians among the philosophical faculty, who insisted that women should not become privatdozenten. In a joint department meeting on the matter, one faculty member protested: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?"Template:Sfn Template:Sfn Hilbert, who believed Noether's qualifications were the only important issue and that the sex of the candidate was irrelevant, objected with indignation and scolded those protesting her habilitation. Although his exact words have not been preserved, his objection is often said to have included the remark that the university was "not a bathhouse."Template:Sfn Template:Sfn Template:Sfn Template:Sfn According to Pavel Alexandrov's recollection, faculty members' opposition to Noether was based not just in sexism, but also in their objections to her social-democratic political beliefs and Jewish ancestry.Template:Sfn

File:Hilbert.jpg

David Hilbert invited Noether to join Göttingen mathematics department in 1915, challenging the views of some of his colleagues that a woman should not teach at a university.

Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death is unknown. At about the same time, Noether's father retired and her brother joined the German Army to serve in World War I. She returned to Erlangen for several weeks, mostly to care for her aging father.Template:Sfn

During her first years teaching at Göttingen, she did not have an official position and was not paid. Her lectures often were advertised under Hilbert's name, and Noether would provide "assistance".Template:Sfn

Soon after arriving at Göttingen, she demonstrated her capabilities by proving the theorem now known as Noether's theorem which shows that a conservation law is associated with any differentiable symmetry of a physical system.Template:Sfn Template:Sfn The paper, Invariante Variationsprobleme, was presented by a colleague, Felix Klein, on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen.Template:Sfn Template:Sfn Noether presumably did not present it herself because she was not a member of the society.Template:Sfn American physicists Leon M. Lederman and Christopher T. Hill argue in their book Symmetry and the Beautiful Universe that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem".Template:Sfn

File:Mathematik Göttingen.jpg

The University of Göttingen allowed Noether's habilitation in 1919, four years after she had begun lecturing at the school.

When World War I ended, the German Revolution of 1918–1919 brought a significant change in social attitudes, including more rights for women. In 1919 the University of Göttingen allowed Noether to proceed with her habilitation (eligibility for tenure). Her oral examination was held in late May, and she successfully delivered her habilitation lecture in June 1919.Template:Sfn Noether became a privatdozent,Template:Sfn and she delivered that fall semester the first lectures listed under her own name.Template:Sfn She was still not paid for her work.Template:Sfn

Three years later, she received a letter from Template:Ill, the Prussian Minister for Science, Art, and Public Education, in which he conferred on her the title of nicht beamteter ausserordentlicher Professor (an untenured professor with limited internal administrative rights and functions).Template:Sfn This was an unpaid "extraordinary" professorship, not the higher "ordinary" professorship, which was a civil-service position. Although it recognized the importance of her work, the position still provided no salary. Noether was not paid for her lectures until she was appointed to the special position of Lehrbeauftragte für Algebra (Lecturer for Algebra) a year later.Template:Sfn Template:Sfn

Work in abstract algebraEdit

Although Noether's theorem had a significant effect upon classical and quantum mechanics, among mathematicians she is best remembered for her contributions to abstract algebra. In his introduction to Noether's Collected Papers, Nathan Jacobson wrote that

The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her — in published papers, in lectures, and in personal influence on her contemporaries.Template:Sfn

Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published a paper about the theory of ideals in which they defined left and right ideals in a ring.Template:Sfn

The following year she published the paper Idealtheorie in Ringbereichen,Template:Sfn analyzing ascending chain conditions with regards to (mathematical) ideals, in which she proved the Lasker–Noether theorem in its full generality. Noted algebraist Irving Kaplansky called this work "revolutionary".Template:Sfn The publication gave rise to the term Noetherian for objects which satisfy the ascending chain condition.Template:Sfn Template:Sfn

File:ETH-BIB-Waerden, Bartel Leendert van der (1903-1996)-Portr 12109.tif

B. L. van der Waerden (pictured in 1980) was heavily influenced by Noether at Göttingen.

In 1924, a young Dutch mathematician, Bartel Leendert van der Waerden, arrived at the University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. Van der Waerden later said that her originality was "absolute beyond comparison".Template:Sfn After returning to Amsterdam, he wrote Moderne Algebra, a central two-volume text in the field; its second volume, published in 1931, borrowed heavily from Noether's work.<ref name="Mactutor Biography"/> Although Noether did not seek recognition, he included as a note in the seventh edition "based in part on lectures by E. Artin and E. Noether".Template:Sfn Template:Sfn Template:Sfn Beginning in 1927, Noether worked closely with Emil Artin, Richard Brauer and Helmut Hasse on noncommutative algebras.Template:Sfn<ref name="Mactutor Biography"/>

Van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which had become a major hub of mathematical and physical research. Russian mathematicians Pavel Alexandrov and Pavel Urysohn were the first of several in 1923.Template:Sfn Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends.Template:Sfn He dubbed her der Noether, using der as an epithet rather than as the masculine German article.Template:Efn Template:Sfn She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship to Princeton University for the 1927–1928 academic year from the Rockefeller Foundation.Template:Sfn Template:Sfn

Graduate studentsEdit

File:EmmyNoether MFO3096.jpg

Noether c. 1930

In Göttingen, Noether supervised more than a dozen doctoral students,<ref name="MacTutorStudents"/> though most were together with Edmund Landau and others as she was not allowed to supervise dissertations on her own.Template:Sfn Template:Sfn Her first was Grete Hermann, who defended her dissertation in February 1925.Template:Sfn Although she is best remembered for her work on the foundations of quantum mechanics, her dissertation was considered an important contribution to ideal theory.Template:Sfn Template:Sfn Hermann later spoke reverently of her "dissertation-mother".Template:Sfn

Around the same time, Heinrich Grell and Rudolf Hölzer wrote their dissertations under Noether, though the latter died of tuberculosis shortly before his defense.Template:Sfn Template:Sfn Template:Sfn Grell defended his thesis in 1926 and went on to work at the University of Jena and the University of Halle, before losing his teaching license in 1935 due to accusations of homosexual acts.<ref name="MacTutorStudents"/> He was later reinstated and became a professor at Humboldt University in 1948.<ref name="MacTutorStudents"/>Template:Sfn

Noether then supervised Werner Weber Template:Sfn and Jakob Levitzki,Template:Sfn who both defended their theses in 1929.Template:Sfn Template:Sfn Weber, who was considered only a modest mathematician,Template:Sfn would later take part in driving Jewish mathematicians out of Göttingen.Template:Sfn Levitzki worked first at Yale University and then at the Hebrew University of Jerusalem in then British-ruled Mandatory Palestine, making significant contributions (in particular Levitzky's theorem and the Hopkins–Levitzki theorem) to ring theory.Template:Sfn

Other Noether Boys included Max Deuring, Hans Fitting, Ernst Witt, Chiungtze C. Tsen and Otto Schilling. Deuring, who had been considered the most promising of Noether's students, was awarded his doctorate in 1930.Template:Sfn Template:Sfn He worked in Hamburg, Marden and GöttingenTemplate:Efn and is known for his contributions to arithmetic geometry.Template:Sfn Fitting graduated in 1931 with a thesis on abelian groupsTemplate:Sfn and is remembered for his work in group theory, particularly Fitting's theorem and the Fitting lemma.Template:Sfn He died at the age of 31 from a bone disease.Template:Sfn

Witt was initially supervised by Noether, but her position was revoked in April 1933 and he was assigned to Gustav Herglotz instead.Template:Sfn He received his PhD in July 1933 with a thesis on the Riemann-Roch theorem and zeta-functions,Template:Sfn and went on to make several contributions that now bear his name.Template:Sfn Tsen, best remembered for proving Tsen's theorem, received his doctorate in December of the same year.Template:Sfn He returned to China in 1935 and started teaching at National Chekiang University,Template:Sfn but died only five years later.Template:Efn Schilling also began studying under Noether, but was forced to find a new advisor due to Noether's emigration. Under Helmut Hasse, he completed his PhD in 1934 at the University of Marburg.Template:Sfn Template:Sfn He later worked as a post doc at Trinity College, Cambridge, before moving to the United States.<ref name="MacTutorStudents"/>

Noether's other students were Wilhelm Dörnte, who received his doctorate in 1927 with a thesis on groups,Template:Sfn Werner Vorbeck, who did so in 1935 with a thesis on splitting fields,<ref name="MacTutorStudents"/> and Wolfgang Wichmann, who did so 1936 with a thesis on p-adic theory.Template:Sfn There is no information about the first two, but it is known that Wichmann supported a student initiative that unsuccessfully attempted to revoke Noether's dismissalTemplate:Sfn and died as a soldier on the Eastern Front during World War II.<ref name="MacTutorStudents"/>

Noether schoolEdit

Noether developed a close circle of mathematicians beyond just her doctoral students who shared Noether's approach to abstract algebra and contributed to the field's development,Template:Sfn a group often referred to as the Noether school.Template:Sfn Template:Sfn An example of this is her close work with Wolfgang Krull, who greatly advanced commutative algebra with his Hauptidealsatz and his dimension theory for commutative rings.Template:Sfn Another is Gottfried Köthe, who contributed to the development of the theory of hypercomplex quantities using Noether and Krull's methods.Template:Sfn

In addition to her mathematical insight, Noether was respected for her consideration of others. Although she sometimes acted rudely toward those who disagreed with her, she nevertheless gained a reputation for constant helpfulness and patient guidance of new students. Her loyalty to mathematical precision caused one colleague to name her "a severe critic", but she combined this demand for accuracy with a nurturing attitude.Template:Sfn In Noether's obituary, Van der Waerden described her as

Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all.Template:Sfn

Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the building was closed for a state holiday, she gathered the class on the steps outside, led them through the woods, and lectured at a local coffee house.Template:Sfn Later, after Nazi Germany dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts.Template:Sfn

Influential lecturesEdit

Noether's frugal lifestyle was at first due to her being denied pay for her work. However, even after the university began paying her a small salary in 1923, she continued to live a simple and modest life. She was paid more generously later in her life, but saved half of her salary to bequeath to her nephew, Gottfried E. Noether.Template:Sfn

Biographers suggest that she was mostly unconcerned about appearance and manners, focusing on her studies. Olga Taussky-Todd, a distinguished algebraist taught by Noether, described a luncheon during which Noether, wholly engrossed in a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped it off from her dress, completely unperturbed".Template:Sfn Appearance-conscious students cringed as she retrieved the handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students once approached her during a break in a two-hour class to express their concern, but they were unable to break through the energetic mathematical discussion she was having with other students.Template:Sfn

Noether did not follow a lesson plan for her lectures.Template:Sfn She spoke quickly and her lectures were considered difficult to follow by many, including Carl Ludwig Siegel and Paul Dubreil.Template:Sfn Template:Sfn Students who disliked her style often felt alienated.Template:Sfn "Outsiders" who occasionally visited Noether's lectures usually spent only half an hour in the room before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been defeated; he has cleared out."Template:Sfn

She used her lectures as a spontaneous discussion time with her students, to think through and clarify important problems in mathematics. Some of her most important results were developed in these lectures, and the lecture notes of her students formed the basis for several important textbooks, such as those of van der Waerden and Deuring.Template:Sfn Noether transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her.Template:Sfn Template:Sfn

Several of her colleagues attended her lectures and she sometimes allowed others (including her students) to receive credit for her ideas, resulting in much of her work appearing in papers not under her name.<ref name="Mactutor Biography">Template:MacTutor</ref>Template:Sfn Noether was recorded as having given at least five semester-long courses at Göttingen:<ref name="scharlau_49">Template:Citation in Template:Harvnb.</ref>

Winter 1924–1925: Gruppentheorie und hyperkomplexe Zahlen [Group Theory and Hypercomplex Numbers]
Winter 1927–1928: Hyperkomplexe Grössen und Darstellungstheorie [Hypercomplex Quantities and Representation Theory]
Summer 1928: Nichtkommutative Algebra [Noncommutative Algebra]
Summer 1929: Nichtkommutative Arithmetik [Noncommutative Arithmetic]
Winter 1929–1930: Algebra der hyperkomplexen Grössen [Algebra of Hypercomplex Quantities]

Moscow State UniversityEdit

File:Paul S Alexandroff 2.jpg

Pavel Alexandrov

In the winter of 1928–1929, Noether accepted an invitation to Moscow State University, where she continued working with P. S. Alexandrov. In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry. She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory.Template:Sfn Template:Sfn Template:Sfn

Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution. She was especially happy to see Soviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from a pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess".Template:Sfn Hermann Weyl recalled that "During the wild times after the Revolution of 1918," Noether "sided more or less with the Social Democrats".Template:Sfn She was from 1919 through 1922 a member of the Independent Social Democrats, a short-lived splinter party. In the words of logician and historian Colin McLarty, "she was not a Bolshevist, but was not afraid to be called one."Template:Sfn

File:Moscow 05-2012 Mokhovaya 05.jpg

Noether taught at Moscow State University in the winter of 1928–1929.

Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933, he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry. Although this effort proved unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union.Template:Sfn

RecognitionEdit

In 1932, Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics.<ref name="Mactutor Biography" /> The prize included a monetary reward of Template:Reichsmark and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to the Göttingen Gesellschaft der Wissenschaften (academy of sciences) and was never promoted to the position of Ordentlicher ProfessorTemplate:Sfn Template:Sfn (full professor).Template:Sfn

Noether's colleagues celebrated her fiftieth birthday, in 1932, in typical mathematicians' style. Helmut Hasse dedicated an article to her in the Mathematische Annalen, wherein he confirmed her suspicion that some aspects of noncommutative algebra are simpler than those of commutative algebra, by proving a noncommutative reciprocity law.Template:Sfn This pleased her immensely. He also sent her a mathematical riddle, which he called the "m_μν-riddle of syllables". She solved it immediately, but the riddle has been lost.Template:Sfn Template:Sfn

File:Internationaler Mathematikerkongress Zürich 1932 - ETH BIB Portr 10680-FL (Johannes Meiner).jpg

Noether visited Zürich in 1932 to deliver a plenary address at the International Congress of Mathematicians.

In September of the same year, Noether delivered a plenary address (großer Vortrag) on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich. The congress was attended by 800 people, including Noether's colleagues Hermann Weyl, Edmund Landau, and Wolfgang Krull. There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.Template:Sfn Template:Sfn

Expulsion from Göttingen by Nazi GermanyEdit

When Adolf Hitler became the German Reichskanzler in January 1933, Nazi activity around the country increased dramatically. At the University of Göttingen, the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by privatdozent and Noether's former student Werner Weber. Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."Template:Sfn

One of the first actions of Hitler's administration was the Law for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" by serving in World War I. In April 1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen."Template:Sfn Template:Sfn Several of Noether's colleagues, including Max Born and Richard Courant, also had their positions revoked.Template:Sfn Template:Sfn

Noether accepted the decision calmly, providing support for others during this difficult time. Hermann Weyl later wrote that "Emmy NoetherTemplate:Sndher courage, her frankness, her unconcern about her own fate, her conciliatory spiritTemplate:Sndwas in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace."Template:Sfn Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory. When one of her students appeared in the uniform of the Nazi paramilitary organization Sturmabteilung (SA), she showed no sign of agitation and, reportedly, even laughed about it later.Template:Sfn Template:Sfn

Refuge at Bryn Mawr and PrincetonEdit

File:Entrance Bryn Mawr.JPG

Bryn Mawr College provided a welcoming home for Noether during the last two years of her life.

As dozens of newly unemployed professors began searching for positions outside of Germany, their colleagues in the United States sought to provide assistance and job opportunities for them. Albert Einstein and Hermann Weyl were appointed by the Institute for Advanced Study in Princeton, while others worked to find a sponsor required for legal immigration. Noether was contacted by representatives of two educational institutions: Bryn Mawr College, in the United States, and Somerville College at the University of Oxford, in England. After a series of negotiations with the Rockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.Template:Sfn Template:Sfn

At Bryn Mawr, Noether met and befriended Anna Wheeler, who had studied at Göttingen just before Noether arrived there. Another source of support at the college was the Bryn Mawr president, Marion Edwards Park, who enthusiastically invited mathematicians in the area to "see Dr. Noether in action!"Template:Sfn Template:Sfn

During her time at Bryn Mawr, Noether formed a group, sometimes called the Noether girls,Template:Sfn of four post-doctoral (Grace Shover Quinn, Marie Johanna Weiss, Olga Taussky-Todd, who all went on to have successful careers in mathematics) and doctoral students (Ruth Stauffer).Template:Sfn They enthusiastically worked through van der Waerden's Moderne Algebra I and parts of Erich Hecke's Theorie der algebraischen Zahlen (Theory of algebraic numbers).Template:Sfn Stauffer was Noether's only doctoral student in the United States, but Noether died shortly before she graduated.Template:Sfn She took her examination with Richard Brauer and received her degree in June 1935,Template:Sfn with a thesis concerning separable normal extensions.Template:Sfn After her doctorate, Stauffer worked as a teacher for a short period and as a statistician for over 30 years.<ref name="MacTutorStudents"/>Template:Sfn

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> She also worked with Abraham Albert and Harry Vandiver.Template:Sfn However, she remarked about Princeton University that she was not welcome at "the men's university, where nothing female is admitted".Template:Sfn

Her time in the United States was pleasant, as she was surrounded by supportive colleagues and absorbed in her favorite subjects.Template:Sfn In the summer of 1934, she briefly returned to Germany to see Emil Artin and her brother Fritz.Template:Sfn The latter, after having been forced out of his job at the Technische Hochschule Breslau, had accepted a position at the Research Institute for Mathematics and Mechanics in Tomsk, in the Siberian Federal District of Russia.Template:Sfn

Although many of her former colleagues had been forced out of the universities, she was able to use the library in Göttingen as a "foreign scholar". Without incident, Noether returned to the United States and her studies at Bryn Mawr.Template:Sfn Template:Sfn

DeathEdit

In April 1935, doctors discovered a tumor in Noether's pelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst "the size of a large cantaloupe".Template:Sfn Two smaller tumors in her uterus appeared to be benign and were not removed to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse on the fourth. On 14 April, Noether fell unconscious, her temperature soared to Template:Convert, and she died. "[I]t is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located." She was 53.Template:Sfn

File:Bryn Mawr College Cloisters.JPG

Noether's ashes were placed under the cloistered walkway of Bryn Mawr's M. Carey Thomas Library.

A few days after Noether's death, her friends and associates at Bryn Mawr held a small memorial service at College President Park's house.Template:Sfn Hermann Weyl and Richard Brauer both traveled from Princeton and delivered eulogies.Template:Sfn In the months that followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, and Pavel Alexandrov in paying their respects.<ref name="einstein"/> Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.Template:Sfn<ref name="APSNews">Template:Cite journal</ref>

Contributions to mathematics and physicsEdit

Noether's work in abstract algebra and topology was influential in mathematics, while Noether's theorem has widespread consequences for theoretical physics and dynamical systems. Noether showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.Template:Sfn Her friend and colleague Hermann Weyl described her scholarly output in three epochs:

(1) the period of relative dependence, 1907–1919
(2) the investigations grouped around the general theory of ideals 1920–1926
(3) the study of the non-commutative algebras, their representations by linear transformations, and their application to the study of commutative number fields and their arithmetics{{#if:Template:Harvnb|{{#if:|}}

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In the first epoch (1907–1919), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. Shortly after moving to Göttingen in 1915, she proved the two Noether's theorems, "one of the most important mathematical theorems ever proved in guiding the development of modern physics".Template:Sfn

In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings.Template:Sfn In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields.Template:Sfn Although the results of Noether's first epoch were impressive and useful, her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B. L. van der Waerden in their obituaries of her.Template:Sfn Template:Sfn

In these epochs, she was not merely applying ideas and methods of the earlier mathematicians; rather, she was crafting new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a completely new theory of ideals in rings, generalizing the earlier work of Richard Dedekind. She is also renowned for developing ascending chain conditionsTemplate:Snda simple finiteness condition that yielded powerful results in her hands.<ref name="ACC">Template:Harvnb. See p. 27: "In 1921, Noether published her famous paper ... [which] dealt with rings whose ideals satisfy the ascending chain condition". See p. 30: "The role of chain conditions in abstract algebra begins with her now classic paper [1921] and culminates with the seminal study [1927]". See p. 28 on strong initial support for her ideas in the 1920s by Pavel Alexandrov and Helmut Hasse, despite "considerable skepticism" from French mathematicians.</ref> Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as the topics of algebraic invariants that had been studied by her father and elimination theory, discussed below.

Historical contextEdit

In the century from 1832 to Noether's death in 1935, the field of mathematics — specifically algebra — underwent a profound revolution whose reverberations are still being felt. Mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g., cubic, quartic, and quintic equations, as well as on the related problem of constructing regular polygons using compass and straightedge. Beginning with Carl Friedrich Gauss's 1832 proof that prime numbers such as five can be factored in Gaussian integers,<ref>Template:Cite journal Reprinted in Template:Cite book</ref> Évariste Galois's introduction of permutation groups in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), William Rowan Hamilton's description of quaternions in 1843, and Arthur Cayley's more modern definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by ever-more-universal rules. Noether's most important contributions to mathematics were to the development of this new field, abstract algebra.Template:Sfn

Background on abstract algebra and begriffliche Mathematik (conceptual mathematics)Edit

Two of the most basic objects in abstract algebra are groups and rings:

A group consists of a set of elements and a single operation which combines a first and a second element and returns a third. The operation must satisfy certain constraints for it to determine a group: it must be closed (when applied to any pair of elements of the associated set, the generated element must also be a member of that set), it must be associative, there must be an identity element (an element which, when combined with another element using the operation, results in the original element, such as by multiplying a number by one), and for every element there must be an inverse element.Template:Sfn Template:Sfn
A ring likewise, has a set of elements, but now has two operations. The first operation must make the set a commutative group, and the second operation is associative and distributive with respect to the first operation. It may or may not be commutative; this means that the result of applying the operation to a first and a second element is the same as to the second and first — the order of the elements does not matter.Template:Sfn If every non-zero element has a multiplicative inverse (an element Template:Math such that Template:Math), the ring is called a division ring. A field is defined as a commutativeTemplate:Efn division ring. For instance, the integers form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication. Any pair of integers can be added or multiplied, always resulting in another integer, and the first operation, addition, is commutative, i.e., for any elements Template:Math and Template:Math in the ring, Template:Math. The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that Template:Math combined with Template:Math might be different from Template:Math combined with Template:Math. Examples of noncommutative rings include matrices and quaternions. The integers do not form a division ring, because the second operation cannot always be inverted; for example, there is no integer Template:Math such that Template:Math.Template:Sfn Template:Sfn

The integers have additional properties which do not generalize to all commutative rings. An important example is the fundamental theorem of arithmetic, which says that every positive integer can be factored uniquely into prime numbers.Template:Sfn Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem, now called the Lasker–Noether theorem, for the ideals of many rings.Template:Sfn As detailed below, Noether's work included determining what properties do hold for all rings, devising novel analogs of the old integer theorems, and determining the minimal set of assumptions required to yield certain properties of rings.

Groups are frequently studied through group representations.Template:Sfn In their most general form, these consist of a choice of group, a set, and an action of the group on the set, that is, an operation which takes an element of the group and an element of the set and returns an element of the set. Most often, the set is a vector space, and the group describes the symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. Rotations are a type of symmetry of space, because the laws of physics themselves do not pick out a preferred direction.Template:Sfn Noether used these sorts of symmetries in her work on invariants in physics.Template:Sfn

A powerful way of studying rings is through their modules. A module consists of a choice of ring, another set, usually distinct from the underlying set of the ring and called the underlying set of the module, an operation on pairs of elements of the underlying set of the module, and an operation which takes an element of the ring and an element of the module and returns an element of the module.Template:Sfn

The underlying set of the module and its operation must form a group. A module is a ring-theoretic version of a group representation: ignoring the second ring operation and the operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring itself. An important special case of this is an algebra.Template:Efn An algebra consists of a choice of two rings and an operation which takes an element from each ring and returns an element of the second ring. This operation makes the second ring into a module over the first.Template:Sfn

Words such as "element" and "combining operation" are very general, and can be applied to many real-world and abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For instance, the elements might be logical propositions, where the first combining operation is exclusive or and the second is logical conjunction.Template:Sfn Theorems of abstract algebra are powerful because they are general; they govern many systems. It might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay Noether's gift to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. In his obituary of Noether, van der Waerden recalled that

The maxim by which Emmy Noether was guided throughout her work might be formulated as follows: "Any relationships between numbers, functions, and operations become transparent, generally applicable, and fully productive only after they have been isolated from their particular objects and been formulated as universally valid concepts."Template:Sfn {{#if:|{{#if:|}}
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This is the begriffliche Mathematik (purely conceptual mathematics) that was characteristic of Noether. This style of mathematics was consequently adopted by other mathematicians, especially in the (then new) field of abstract algebra.Template:Sfn

First epoch (1908–1919)Edit

Algebraic invariant theoryEdit

File:Emmy Noether - Table of invariants 2.jpg

Table 2 from Noether's dissertationTemplate:Sfn on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables x and u. The horizontal direction of the table lists the invariants with increasing grades in x, while the vertical direction lists them with increasing grades in u.

Much of Noether's work in the first epoch of her career was associated with invariant theory, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations.Template:Sfn As an everyday example, if a rigid metre-stick is rotated, the coordinates of its endpoints change, but its length remains the same. A more sophisticated example of an invariant is the discriminant Template:Math of a homogeneous quadratic polynomial Template:Math, where Template:Mvar and Template:Mvar are indeterminates. The discriminant is called "invariant" because it is not changed by linear substitutions Template:Math and Template:Math with determinant Template:Math. These substitutions form the special linear group Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

One can ask for all polynomials in Template:Mvar, Template:Mvar, and Template:Mvar that are unchanged by the action of Template:Math; these turn out to be the polynomials in the discriminant.Template:Sfn More generally, one can ask for the invariants of homogeneous polynomials Template:Math of higher degree, which will be certain polynomials in the coefficients Template:Math, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.Template:Sfn

One of the main goals of invariant theory was to solve the "finite basis problem". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called generators, and then, adding or multiplying the generators together.Template:Sfn For example, the discriminant gives a finite basis (with one element) for the invariants of a quadratic polynomial.Template:Sfn

Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables.Template:Sfn Template:Sfn He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables.Template:Sfn Template:Sfn Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.Template:Sfn

Noether followed Gordan's lead, writing her doctoral dissertation and several other publications on invariant theory. She extended Gordan's results and also built upon Hilbert's research. Later, she would disparage this work, finding it of little interest and admitting to forgetting the details of it.Template:Sfn Hermann Weyl wrote,

[A] greater contrast is hardly imaginable than 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.Template:Sfn

Galois theoryEdit

Galois theory concerns transformations of number fields that permute the roots of an equation.<ref>Template:Harvnb</ref> Consider a polynomial equation of a variable Template:Math of degree Template:Math, in which the coefficients are drawn from some ground field, which might be, for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not be choices of Template:Math, which make this polynomial evaluate to zero. Such choices, if they exist, are called roots.Template:Sfn For example, if the polynomial is Template:Math and the field is the real numbers, then the polynomial has no roots, because any choice of Template:Math makes the polynomial greater than or equal to one.Template:Sfn If the field is extended, however, then the polynomial may gain roots,Template:Sfn and if it is extended enough, then it always has a number of roots equal to its degree.Template:Sfn

Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, Template:Math and Template:Math, where Template:Math is the imaginary unit, that is, Template:Math More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.<ref>Template:Harvnb</ref>

The Galois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial.<ref>Template:Harvnb</ref> (These transformations are called automorphisms.) The Galois group of Template:Nowrap consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends Template:Math to Template:Math. Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a permutation of the Template:Math roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups of the Galois group.<ref>Template:Harvnb</ref>

In 1918, Noether published a paper on the inverse Galois problem.<ref>Template:Harvnb.</ref> Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "Noether's problem", which asks whether the fixed field of a subgroup G of the permutation group Template:Math acting on the field Template:Math always is a pure transcendental extension of the field Template:Math. (She first mentioned this problem in a 1913 paper,<ref>Template:Harvnb.</ref> where she attributed the problem to her colleague Fischer.) She showed this was true for Template:Math = 2, 3, or 4. In 1969, Richard Swan found a counter-example to Noether's problem, with Template:Math = 47 and Template:Math a cyclic group of order 47<ref>Template:Harvnb.</ref> (although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved.<ref>Template:Harvnb.</ref>

PhysicsEdit

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Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity,Template:Sfn a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, in a 1918 paper.<ref>Template:Harvnb</ref> This paper presented two theorems, of which the first is known as Noether's theorem.<ref>Template:Harvnb</ref> Together, these theorems not only solve the problem for general relativity, but also determine the conserved quantities for every system of physical laws that possesses some continuous symmetry.<ref>{{#invoke:citation/CS1|citation |CitationClass=web

}}</ref> Upon receiving her work, Einstein wrote to Hilbert:<templatestyles src="Template:Blockquote/styles.css" />

Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.<ref>Template:Harvnb</ref>{{#if:|{{#if:|}}
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For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved.Template:Sfn<ref name="ledhill">Template:Harvnb.</ref> The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the system is responsible for the conservation law. As another example, if a physical experiment works the same way at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.Template:Sfn<ref>Template:Cite book</ref>

At the time, physicists were not familiar with Sophus Lie's theory of continuous groups, on which Noether had built. Many physicists first learned of Noether's theorem from an article by Edward Lee Hill that presented only a special case of it. Consequently, the full scope of her result was not immediately appreciated.<ref>Template:Harvnb</ref> During the latter half of the 20th century, however, Noether's theorem became a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon: If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.<ref name="neeman_1999" />

Second epoch (1920–1926)Edit

Ascending and descending chain conditionsEdit

In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions.<ref name="ACC"/> A sequence of non-empty subsets Template:Math, ... of a set Template:Math is usually said to be ascending if each is a subset of the next:

<math>A_{1} \subseteq A_{2} \subseteq A_{3} \subseteq \cdots.</math>

Conversely, a sequence of subsets of Template:Math is called descending if each contains the next subset:

<math>A_{1} \supseteq A_{2} \supseteq A_{3} \supseteq \cdots.</math>

A chain becomes constant after a finite number of steps if there is an Template:Math such that <math>A_n = A_m</math> for all Template:Math. A collection of subsets of a given set satisfies the ascending chain condition if every ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.Template:Sfn Chain conditions can be used to show that every set of sub-objects has a maximal/minimal element, or that a complex object can be generated by a smaller number of elements.Template:Sfn

Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called Noetherian in her honor.Template:Sfn By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. A Noetherian module is a module in which every strictly ascending chain of submodules becomes constant after a finite number of steps.Template:Sfn Template:Sfn A Noetherian space is a topological space whose open subsets satisfy the ascending chain condition;Template:Efn this definition makes the spectrum of a Noetherian ring a Noetherian topological space.Template:Sfn Template:Sfn

The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are Noetherian; and, mutatis mutandis, the same holds for submodules and quotient modules of a Noetherian module.Template:Sfn The chain condition also may be inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.Template:Sfn

Another application of such chain conditions is in Noetherian induction Template:Sndalso known as well-founded induction Template:Sndwhich is a generalization of mathematical induction. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose that Template:Math is a partially ordered set. One way of proving a statement about the objects of Template:Math is to assume the existence of a counterexample and deduce a contradiction, thereby proving the contrapositive of the original statement. The basic premise of Noetherian induction is that every non-empty subset of Template:Math contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the minimal counterexample. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Commutative rings, ideals, and modulesEdit

Noether's paper, Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921),Template:Sfn is the foundation of general commutative ring theory, and gives one of the first general definitions of a commutative ring.Template:Efn Template:Sfn Before her paper, most results in commutative algebra were restricted to special examples of commutative rings, such as polynomial rings over fields or rings of algebraic integers. Noether proved that in a ring which satisfies the ascending chain condition on ideals, every ideal is finitely generated. In 1943, French mathematician Claude Chevalley coined the term Noetherian ring to describe this property.Template:Sfn A major result in Noether's 1921 paper is the Lasker–Noether theorem, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings.Template:Sfn Template:Sfn The Lasker–Noether theorem can be viewed as a generalization of the fundamental theorem of arithmetic which states that any positive integer can be expressed as a product of prime numbers, and that this decomposition is unique.Template:Sfn

Noether's work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields, 1927)<ref>Template:Harvnb.</ref> characterized the rings in which the ideals have unique factorization into prime ideals (now called Dedekind domains).Template:Sfn Noether showed that these rings were characterized by five conditions: they must satisfy the ascending and descending chain conditions, they must possess a unit element, but no zero divisors, and they must be integrally closed in their associated field of fractions.Template:Sfn Template:Sfn This paper also contains what now are called the isomorphism theorems,Template:Sfn which describe some fundamental natural isomorphisms, and some other basic results on Noetherian and Artinian modules.Template:Sfn

Elimination theoryEdit

In 1923–1924, Noether applied her ideal theory to elimination theory in a formulation that she attributed to her student, Kurt Hentzelt. She showed that fundamental theorems about the factorization of polynomials could be carried over directly.Template:Sfn Template:Sfn Template:Sfn

Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, often by the method of resultants.Template:Sfn For illustration, a system of equations often can be written in the form

Template:Math

where a matrix (or linear transform) Template:Math (without the variable Template:Math) times a vector Template:Math (that only has non-zero powers of Template:Math) is equal to the zero vector, Template:Math. Hence, the determinant of the matrix Template:Math must be zero, providing a new equation in which the variable Template:Math has been eliminated.

Invariant theory of finite groupsEdit

Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper,Template:Sfn Noether found a solution to the finite basis problem for a finite group of transformations Template:Math acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is called Noether's bound. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field is coprime to <math>\left|G\right|!</math> (the factorial of the order <math>\left|G\right|</math> of the group Template:Math). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number <math>\left|G\right|</math>,Template:Sfn but Noether was not able to determine whether this bound was correct when the characteristic of the field divides <math>\left|G\right|!</math> but not <math>\left|G\right|</math>. For many years, determining the truth or falsehood of this bound for this particular case was an open problem, called "Noether's gap". It was finally solved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.Template:Sfn Template:Sfn

In her 1926 paper,Template:Sfn Noether extended Hilbert's theorem to representations of a finite group over any field; the new case that did not follow from Hilbert's work is when the characteristic of the field divides the order of the group. Noether's result was later extended by William Haboush to all reductive groups by his proof of the Mumford conjecture.Template:Sfn In this paper Noether also introduced the Noether normalization lemma, showing that a finitely generated domain Template:Math over a field Template:Math has a set Template:Math} of algebraically independent elements such that Template:Math is integral over Template:Math.

TopologyEdit

File:Mug and Torus morph.gif

A continuous deformation (homotopy) of a coffee cup into a doughnut (torus) and back

As noted by Hermann Weyl in his obituary, Noether's contributions to topology illustrate her generosity with ideas and how her insights could transform entire fields of mathematics.Template:Sfn In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as their connectedness. An old joke is that "a topologist cannot distinguish a donut from a coffee mug", since they can be continuously deformed into one another.Template:Sfn

Noether is credited with fundamental ideas that led to the development of algebraic topology from the earlier combinatorial topology, specifically, the idea of homology groups.Template:Sfn According to Alexandrov, Noether attended lectures given by him and Heinz Hopf in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle"Template:Sfn and he continues that,

When ... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly the groups of algebraic complexes and cycles of a given polyhedron and the subgroup of the cycle group consisting of cycles homologous to zero; instead of the usual definition of Betti numbers, she suggested immediately defining the Betti group as the complementary (quotient) group of the group of all cycles by the subgroup of cycles homologous to zero. This observation now seems self-evident, but in those years (1925–1928) this was a completely new point of view.Template:Sfn{{#if:|{{#if:|}}
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Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others,Template:Sfn and it became a frequent topic of discussion among the mathematicians of Göttingen.<ref>Template:Citation in Template:Harvnb</ref> Noether observed that her idea of a Betti group makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subjectTemplate:Sfn "bears the imprint of these remarks of Emmy Noether".Template:Sfn Noether mentions her own topology ideas only as an aside in a 1926 publication,Template:Sfn where she cites it as an application of group theory.<ref>Template:Citation in Template:Harvnb</ref>

This algebraic approach to topology was also developed independently in Austria. In a 1926–1927 course given in Vienna, Leopold Vietoris defined a homology group, which was developed by Walther Mayer into an axiomatic definition in 1928.<ref>Template:Citation in Template:Harvnb</ref>

File:Helmut Hasse.jpg

Helmut Hasse worked with Noether and others to found the theory of central simple algebras.

Third epoch (1927–1935)Edit

Hypercomplex numbers and representation theoryEdit

Much work on hypercomplex numbers and group representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these earlier results and gave the first general representation theory of groups and algebras.Template:Sfn Template:Sfn This single work by Noether was said to have ushered in a new period in modern algebra and to have been of fundamental importance for its development.Template:Sfn

Briefly, Noether subsumed the structure theory of associative algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings satisfying ascending chain conditions.Template:Sfn

Noncommutative algebraEdit

Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the theory of central simple algebras.Template:Sfn

A paper by Noether, Helmut Hasse, and Richard Brauer pertains to division algebras,<ref name = "hasse_1932">Template:Harvnb.</ref> which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a finite-dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is trivial), and from this, deduced their Hauptsatz ("main theorem"):

Every finite-dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension.

These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra Template:Math are splitting fields.Template:Sfn This paper also contains the Skolem–Noether theorem, which states that any two embeddings of an extension of a field Template:Math into a finite-dimensional central simple algebra over Template:Math are conjugate. The Brauer–Noether theorem Template:Sfn gives a characterization of the splitting fields of a central division algebra over a field.Template:Sfn

LegacyEdit

File:Emmy-noether-campus siegen.jpg

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File:Ruhmeshalle Muenchen Emmy Noether Mathematikerin-1 retusche.jpg

Noether is also one of the carved stone busts displayed in Germany's Ruhmeshalle München (or Munich Hall of Fame in English)

Noether's work continues to be relevant for the development of theoretical physics and mathematics, and she is considered one of the most important mathematicians of the twentieth century.<ref name="APSNews"/><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> During her lifetime and even until today, Noether has also been characterized as the greatest woman mathematician in recorded historyTemplate:Sfn Template:Sfn Template:Sfn by mathematicians such as Pavel Alexandrov,Template:Sfn Hermann Weyl,Template:Sfn and Jean Dieudonné.Template:Sfn

In a letter to The New York Times, Albert Einstein wrote:<ref name="einstein" />

In the judgment of the most competent living mathematicians, Fräulein 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.{{#if:|{{#if:|}}
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In his obituary, fellow algebraist B. L. van der Waerden says that her mathematical originality was "absolute beyond comparison",Template:Sfn and Hermann Weyl said that Noether "changed the face of [abstract] algebra by her work".<ref name="weyl_128">Template:Harvnb</ref> Mathematician and historian Jeremy Gray wrote that any textbook on abstract algebra bears the evidence of Noether's contributions: "Mathematicians simply do ring theory her way."Template:Sfn Several things now bear her name, including many mathematical objects,Template:Sfn and an asteroid, 7001 Noether.Template:Sfn