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**Amalie Emmy Noether**^{[a]} (US: /ˈnʌtər/, UK: /ˈnɜːtə/; German: [ˈnøːtɐ]; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics.^{[4]} She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics.^{[5]}^{[6]}^{[7]} 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.^{[8]}

Emmy Noether | |
---|---|

Born | Amalie Emmy Noether 23 March 1882 |

Died | 14 April 1935 | (aged 53)

Nationality | German |

Alma mater | University of Erlangen |

Known for | |

Awards | Ackermann–Teubner Memorial Award (1932) |

Scientific career | |

Fields | Mathematics and physics |

Institutions | |

Thesis | On Complete Systems of Invariants for Ternary Biquadratic Forms (1907) |

Doctoral advisor | Paul Gordan |

Doctoral students |

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, where her father lectured. After completing her doctorate in 1907^{[9]} under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. 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*.^{[9]}

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, Ruth Stauffer, Grace Shover Quinn, and Olga Taussky-Todd. At the same time, she lectured and performed research at the Institute for Advanced Study in Princeton, New Jersey.^{[9]}

Noether's mathematical work has been divided into three "epochs".^{[10]} 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".^{[11]} In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra".^{[12]} 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.

Emmy Noether was born on 23 March 1882. She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from Jewish merchant families.^{[13]} Her first name was "Amalie", but she began using her middle name at a young age and she invariably used the name "Emmy Noether" in her adult life and her publications.^{[a]}

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.^{[14]} 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.^{[15]}

She 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.^{[16]} Fritz Noether was born in 1884, studied in Munich and made contributions to applied mathematics. He was executed in the Soviet Union in 1941.^{[17]} 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.^{[18]}^{[19]}

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,^{[20]} at which her father was a professor.^{[21]}

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".^{[22]} 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.^{[20]}^{[23]}^{[24]}

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.^{[25]}

In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded.^{[26]} 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.^{[27]} 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*),^{[28]} in 1907, graduating *summa cum laude* later that year.^{[29]} 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.^{[30]}^{[31]} 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.^{[31]}^{[32]}

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.^{[33]} In 1910 and 1911 she published an extension of her thesis work from three variables to *n* variables.^{[34]}

Gordan retired in 1910,^{[35]} and Noether taught under his successors, Erhard Schmidt and Ernst Fischer, who took over from the former in 1911.^{[36]} 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.^{[37]}^{[38]} 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.^{[39]}^{[40]}

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.^{[41]} This phase marked Noether's first exposure to abstract algebra, the field to which she would make groundbreaking contributions.^{[42]}

In Erlangen, Noether advised two doctoral students:^{[43]} Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916.^{[44]}^{[45]} 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^{[46]} while Seidelmann became a professor in Munich.^{[43]}

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?"^{[47]}^{[48]} 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."^{[37]}^{[47]}^{[49]}^{[50]} 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.^{[50]}

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.^{[51]}

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".^{[52]}

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.^{[48]}^{[53]} 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.^{[54]}^{[55]} Noether presumably did not present it herself because she was not a member of the society.^{[56]} 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".^{[11]}

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.^{[57]} Noether became a *privatdozent*,^{[58]} and she delivered that fall semester the first lectures listed under her own name.^{[59]} She was still not paid for her work.^{[52]}

Three years later, she received a letter from Otto BoelitzPrussian 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).^{[60]} 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* a year later.^{[61]}^{[62]}

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.

^{[1]}

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.^{[42]}

The following year she published the paper *Idealtheorie in Ringbereichen*,^{[63]} 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".^{[64]} The publication gave rise to the term *Noetherian* for objects which satisfy the ascending chain condition.^{[64]}^{[65]}

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".^{[66]} 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.^{[67]} 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".^{[68]}^{[69]}^{[70]} Beginning in 1927, Noether worked closely with Emil Artin, Richard Brauer and Helmut Hasse on noncommutative algebras.^{[37]}^{[67]}

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.^{[71]} Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends.^{[72]} He began referring to her as *der Noether*, using the masculine German article as a term of endearment to show his respect. 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.^{[72]}^{[73]}

In Göttingen, Noether supervised more than a dozen doctoral students,^{[43]} though most were together with Edmund Landau and others as she was not allowed to supervise dissertations on her own.^{[74]}^{[75]} Her first was Grete Hermann, who defended her dissertation in February 1925.^{[76]} Although best remembered for her work on the foundations of quantum mechanics, her dissertation was considered an important contribution to ideal theory.^{[77]}^{[78]} Hermann later spoke reverently of her "dissertation-mother".^{[76]}

Around the same time, Heinrich Grell and Rudolf Hölzer wrote their dissertations under Noether, though the latter died of tuberculosis shortly before his defense.^{[76]}^{[79]}^{[80]} 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.^{[43]} He was later reinstated and became a professor at Humboldt University in 1948.^{[43]}^{[76]}

Noether then supervised Werner Weber^{[81]} and Jakob Levitzki,^{[82]} who both defended their theses in 1929.^{[83]}^{[84]} Weber, who was considered only a modest mathematician,^{[74]} would later take part in driving Jewish mathematicians out of Göttingen.^{[85]} Levitzki worked first at Yale University and then at the Hebrew University of Jerusalem in Palestine, making significant contributions (in particular Levitzky's theorem and the Hopkins–Levitzki theorem) to ring theory.^{[84]}

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.^{[86]}^{[87]} He worked in Hamburg, Marden and Göttingen^{[b]} and is known for his contributions to arithmetic geometry.^{[89]} Fitting graduated in 1931 with a thesis on abelian groups^{[90]} and is remembered for his work in group theory, particularly Fitting's theorem and the Fitting lemma.^{[91]} He died at the age of 31 from a bone disease.^{[92]}

Witt was initially supervised by Noether, but her position was revoked in April 1933 and he was assigned to Gustav Herglotz instead.^{[92]} He received his PhD in July 1933 with a thesis on the Riemann-Roch theorem and zeta-functions,^{[93]} and went on to make several contributions that now bear his name.^{[91]} Tsen, best remembered for proving Tsen's theorem, received his doctorate in December of the same year.^{[94]} He returned to China in 1935 and started teaching at National Chekiang University,^{[91]} but died only five years later.^{[c]} 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.^{[91]}^{[96]} He later worked as a post doc at Trinity College, Cambridge, before moving to the United States.^{[43]}

Noether's other students were Wilhelm Dörnte, who received his doctorate in 1927 with a thesis on groups,^{[97]} Werner Vorbeck, who did so in 1935 with a thesis on splitting fields,^{[43]} and Wolfgang Wichmann, who did so 1936 with a thesis on p-adic theory.^{[98]} 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 dismissal^{[99]} and died as a soldier on the Eastern Front during World War II.^{[43]}

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,^{[100]} a group often referred to as the *Noether school*.^{[101]}^{[102]} 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.^{[103]} Another is Gottfried Köthe, who contributed to the development of the theory of hypercomplex quantities using Noether and Krull's methods.^{[103]}

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.^{[104]} 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.

^{[66]}

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.^{[105]} Later, after Nazi Germany dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts.^{[106]}

Noether's frugal lifestyle at first was 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.^{[107]}

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".^{[108]} 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.^{[109]}

Noether did not follow a lesson plan for her lectures.^{[66]} She spoke quickly and her lectures were considered difficult to follow by many, including Carl Ludwig Siegel and Paul Dubreil.^{[110]}^{[111]} Students who disliked her style often felt alienated.^{[112]} "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."^{[113]}

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.^{[66]} Noether transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her.^{[114]}^{[115]}

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.^{[67]}^{[68]} Noether was recorded as having given at least five semester-long courses at Göttingen:^{[116]}

- 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*]

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.^{[117]}^{[118]}^{[119]}

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".^{[120]} Hermann Weyl recalled that "During the wild times after the Revolution of 1918," Noether "sided more or less with the Social Democrats".^{[37]} 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."^{[121]}

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.^{[120]}

In 1932 Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics.^{[67]} The prize included a monetary reward of 500 ℛ︁ℳ︁ 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 Professor*^{[122]}^{[123]} (full professor).^{[60]}

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.^{[124]} 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.^{[122]}^{[123]}

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.^{[123]}^{[125]}

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."^{[85]}

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."^{[126]}^{[127]} Several of Noether's colleagues, including Max Born and Richard Courant, also had their positions revoked.^{[126]}^{[127]}

Noether accepted the decision calmly, providing support for others during this difficult time. Hermann Weyl later wrote that "Emmy Noether – her courage, her frankness, her unconcern about her own fate, her conciliatory spirit – was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace."^{[85]} 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.^{[126]}^{[127]}

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.^{[128]}^{[129]}

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!"^{[130]}^{[131]}

During her time at Bryn Mawr, Noether formed a group, sometimes called the *Noether girls,*^{[132]} 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).^{[133]} 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*).^{[134]} Stauffer was Noether's only doctoral student in the United States, but Noether died shortly before she graduated.^{[135]} She took her examination with Richard Brauer and received her degree in June 1935,^{[136]} with a thesis concerning separable normal extensions.^{[137]} After her doctorate, Stauffer worked as a teacher for a short period and as a statistician for over 30 years.^{[43]}^{[136]}

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen.^{[138]} She also worked with Abraham Albert and Harry Vandiver.^{[139]} However, she remarked about Princeton University that she was not welcome at "the men's university, where nothing female is admitted".^{[140]}

Her time in the United States was pleasant, as she was surrounded by supportive colleagues and absorbed in her favorite subjects.^{[141]} In the summer of 1934, she briefly returned to Germany to see Emil Artin and her brother Fritz.^{[142]} 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.^{[142]} He was subsequently executed during the Medvedev Forest massacre.^{[143]}

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.^{[144]}^{[145]}

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".^{[146]} 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 109 °F (42.8 °C), 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.^{[146]}

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.^{[147]} Hermann Weyl and Richard Brauer both traveled from Princeton and delivered eulogies.^{[148]} 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.^{[5]} Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.^{[149]}^{[150]}

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.^{[39]} 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

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".^{[11]}

In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings.^{[151]} In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields.^{[152]} 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.^{[37]}^{[66]}

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 conditions – a simple finiteness condition that yielded powerful results in her hands.^{[153]} 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.

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,^{[154]} 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.^{[155]}

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.^{[156]}^{[157]} - 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.^{[158]}If every non-zero element has a multiplicative inverse (an element*x*such that*ax*=*xa*= 1), the ring is called a*division ring*. A*field*is defined as a commutative^{[d]}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*a*and*b*in the ring,*a*+*b*=*b*+*a*. The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that*a*combined with*b*might be different from*b*combined with*a*. 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*a*such that 3*a*= 1.^{[159]}^{[160]}

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.^{[161]} 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.^{[162]} Much of Noether's work lay in determining what properties *do* hold for all rings, in devising novel analogs of the old integer theorems, and in determining the minimal set of assumptions required to yield certain properties of rings.^{[citation needed]}

Groups are frequently studied through *group representations*.^{[163]} 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.^{[164]} Noether used these sorts of symmetries in her work on invariants in physics.^{[165]}

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.^{[166]}

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*.^{[e]} 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. Often the first ring is a field.^{[citation needed]}

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.^{[167]} 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."

^{[168]}

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.^{[169]}

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. As an everyday example, if a rigid yardstick is rotated, the coordinates of its endpoints change, but its length remains the same. Invariant theory was an active area of research in the later nineteenth century, prompted in part by Felix Klein's Erlangen program, according to which different types of geometry should be characterized by their invariants under transformations, e.g., the cross-ratio of projective geometry.^{[citation needed]}

An example of an *invariant* is the discriminant *B*^{2} − 4 *AC* of a binary quadratic form **x·**A **x** + **y·**B **x** + **y·**C **y**, where **x** and **y** are vectors and "**·**" is the dot product or "inner product" for the vectors. A, B, and C are linear operators on the vectors – typically matrices. The discriminant is called "invariant" because it is not changed by linear substitutions * x* →

One can ask for all polynomials in A, B, and C that are unchanged by the action of *SL*_{2}; these are called the invariants of binary quadratic forms and turn out to be the polynomials in the discriminant. More generally, one can ask for the invariants of homogeneous polynomials A_{0} *x*^{r} *y*^{0} + ... + A_{r} x^{0} *y*^{r} of higher degree, which will be certain polynomials in the coefficients A_{0}, ..., A_{r}, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.^{[citation needed]}

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. For example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.^{[citation needed]}

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.^{[170]}^{[171]} 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.^{[172]}^{[173]} Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.^{[174]}

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.^{[175]} 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.

^{[176]}

Galois theory concerns transformations of number fields that permute the roots of an equation.^{[177]} Consider a polynomial equation of a variable *x* of degree *n*, 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 *x*, which make this polynomial evaluate to zero. Such choices, if they exist, are called roots.^{[178]} For example, if the polynomial is *x*^{2} + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of *x* makes the polynomial greater than or equal to one.^{[179]} If the field is extended, however, then the polynomial may gain roots,^{[180]} and if it is extended enough, then it always has a number of roots equal to its degree.^{[181]}

Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, +*i* and −*i*, where *i* is the imaginary unit, that is, *i*^{ 2} = −1. More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.^{[182]}

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.^{[183]} (These transformations are called automorphisms.) The Galois group of *x*^{2} + 1 consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends +*i* to −*i*. 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 *n* 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.^{[184]}

In 1918, Noether published a paper on the inverse Galois problem.^{[185]} 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 *S*_{n} acting on the field *k*(*x*_{1}, ..., *x*_{n}) always is a pure transcendental extension of the field *k*. (She first mentioned this problem in a 1913 paper,^{[186]} where she attributed the problem to her colleague Fischer.) She showed this was true for *n* = 2, 3, or 4. In 1969, Richard Swan found a counter-example to Noether's problem, with *n* = 47 and *G* a cyclic group of order 47^{[187]} (although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved.^{[188]}

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,^{[189]} 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.^{[190]} This paper presented two theorems, of which the first is known as Noether's theorem.^{[191]} 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.^{[192]} Upon receiving her work, Einstein wrote to Hilbert:

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.

^{[193]}

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.^{[165]}^{[194]} 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 has the same outcome 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.^{[195]}

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.^{[196]} 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.^{[8]}

In this epoch, Noether became famous for her deft use of ascending (*Teilerkettensatz*) or descending (*Vielfachenkettensatz*) chain conditions.^{[153]} A sequence of non-empty subsets *A*_{1}, *A*_{2}, *A*_{3}, ... of a set *S* is usually said to be *ascending*, if each is a subset of the next

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

A chain *becomes constant after a finite number of steps* if there is an *n* such that for all *m* ≥ *n*. A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps.^{[197]} It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.^{[198]}

Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects – and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage. For example, 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. These conclusions often are crucial steps in a proof.^{[citation needed]}

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.^{[199]} 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.^{[200]} A Noetherian space is a topological space whose open subsets satisfy the ascending chain condition;^{[g]} this definition makes the spectrum of a Noetherian ring a Noetherian topological space.^{[201]}^{[202]}

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 likewise, Noetherian; and, *mutatis mutandis*, the same holds for submodules and quotient modules of a Noetherian module. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. 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.^{[citation needed]}

Another application of such chain conditions is in Noetherian induction – also known as well-founded induction – which 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 *S* is a partially ordered set. One way of proving a statement about the objects of *S* 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 *S* 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.^{[citation needed]}

Noether's paper, *Idealtheorie in Ringbereichen* (*Theory of Ideals in Ring Domains*, 1921),^{[63]} is the foundation of general commutative ring theory, and gives one of the first general definitions of a commutative ring.^{[203]} 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.^{[203]} 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. 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.^{[204]}

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)^{[205]} characterized the rings in which the ideals have unique factorization into prime ideals as the Dedekind domains: integral domains that are Noetherian, 0- or 1-dimensional, and integrally closed in their quotient fields. This paper also contains what now are called the isomorphism theorems, which describe some fundamental natural isomorphisms, and some other basic results on Noetherian and Artinian modules.^{[citation needed]}

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.^{[206]}^{[207]}^{[208]} Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, often by the method of resultants.^{[209]}

For illustration, a system of equations often can be written in the form

- Mv = 0

where a matrix (or linear transform) M (without the variable x) times a vector v (that only has non-zero powers of x) is equal to the zero vector, 0. Hence, the determinant of the matrix M must be zero, providing a new equation in which the variable x has been eliminated.

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,^{[210]} Noether found a solution to the finite basis problem for a finite group of transformations *G* 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 (the factorial of the order of the group *G*). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number ,^{[211]} but Noether was not able to determine whether this bound was correct when the characteristic of the field divides but not . 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.^{[212]}^{[213]}

In her 1926 paper,^{[214]} 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.^{[215]} In this paper Noether also introduced the *Noether normalization lemma*, showing that a finitely generated domain *A* over a field *k* has a set {*x*_{1}, ..., *x*_{n}} of algebraically independent elements such that *A* is integral over *k*[*x*_{1}, ..., *x*_{n}].

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.^{[37]} 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.^{[216]}

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.^{[217]} 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"^{[218]} 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.

^{[219]}

Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others,^{[219]} and it became a frequent topic of discussion among the mathematicians of Göttingen.^{[220]} Noether observed that her idea of a Betti group makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subject^{[221]} "bears the imprint of these remarks of Emmy Noether".^{[222]} Noether mentions her own topology ideas only as an aside in a 1926 publication,^{[223]} where she cites it as an application of group theory.^{[224]}

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.^{[225]}

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.^{[226]}^{[227]} 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.^{[228]}

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.^{[227]}

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.^{[229]}

A paper by Noether, Helmut Hasse, and Richard Brauer pertains to division algebras,^{[230]} 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 *D* are splitting fields.^{[231]} This paper also contains the Skolem–Noether theorem, which states that any two embeddings of an extension of a field *k* into a finite-dimensional central simple algebra over *k* are conjugate. The Brauer–Noether theorem^{[232]} gives a characterization of the splitting fields of a central division algebra over a field.^{[233]}

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.^{[150]}^{[234]}

During her lifetime and even until today, Noether has also been characterized as the greatest woman mathematician in recorded history^{[6]}^{[7]}^{[235]} by mathematicians such as Pavel Alexandrov,^{[236]} Hermann Weyl,^{[237]} and Jean Dieudonné.^{[238]}

In a letter to *The New York Times*, Albert Einstein wrote:^{[5]}

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.

In his obituary, fellow algebraist B.L. van der Waerden says that her mathematical originality was "absolute beyond comparison",^{[239]} and Hermann Weyl said that Noether "changed the face of [abstract] algebra by her work".^{[12]} 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."^{[199]} Several things now bear her name, including many mathematical objects,^{[199]} and an asteroid, 7001 Noether.^{[240]}

- ^
^{a}^{b}Emmy is the*Rufname*, the second of two official given names, intended for daily use. This can be seen in the résumé submitted by Noether to the University of Erlangen in 1907.^{[1]}^{[2]}Sometimes*Emmy*is mistakenly reported as a short form for*Amalie*, or misreported as*Emily*; the latter was used by Lee Smolin in a letter for The Reality Club.^{[3]} **^**When Noether was forced to leave Germany in 1933, she wished for the university to appoint Deuring as her successor,^{[88]}but he only started teaching there in 1950.^{[87]}**^**Accounts of Tsen's date of death vary: Kimberling (1981, p. 41) states that he died "some time in 1939 or 40" and Ding, Kang & Tan (1999) state that he died in November 1940, but a local newspaper recorded his date of death as 1 October 1940.^{[95]}**^**The nomenclature is not consistent.**^**The word*algebra*means both a subject within mathematics as well as an object studied in the subject of algebra.**^**There are no invariants under the general linear group of all invertible linear transformations because these transformations can be multiplication by a scaling factor. To remedy this, classical invariant theory also considered*relative invariants*, which were forms invariant up to a scale factor.**^**Or whose closed subsets satisfy the descending chain condition.^{[201]}

- ^
^{a}^{b}Noether 1983. **^**Tollmien, Cordula. "Emmy Noether (1882–1935) – Lebensläufe".*physikerinnen.de*. Archived from the original on 29 September 2007. Retrieved 13 April 2024.**^**Smolin, Lee (21 March 1999). "Lee Smolin on 'Special Relativity: Why Cant You Go Faster Than Light?' by W. Daniel Hillis; Hillis Responds".*Edge.org*. Edge Foundation, Inc. Archived from the original on 30 July 2012. Retrieved 6 March 2012.But I think very few non-experts will have heard either of it or its maker – Emily Noether, a great German mathematician. ... This also requires Emily Noether's insight, that conserved quantities have to do with symmetries of natural law.

**^**Conover, Emily (12 June 2018). "In her short life, mathematician Emmy Noether changed the face of physics".*Science News*. Archived from the original on 26 March 2023. Retrieved 2 July 2018.- ^
^{a}^{b}^{c}Einstein, Albert (1 May 1935), "Professor Einstein Writes in Appreciation of a Fellow-Mathematician",*The New York Times*(published 4 May 1935), retrieved 13 April 2008. Transcribed online at the MacTutor History of Mathematics Archive. - ^
^{a}^{b}Alexandrov 1981, p. 100. - ^
^{a}^{b}Kimberling 1982. - ^
^{a}^{b}Ne'eman, Yuval,*The Impact of Emmy Noether's Theorems on XXIst Century Physics*in Teicher 1999, pp. 83–101. - ^
^{a}^{b}^{c}Ogilvie & Harvey 2000, p. 949. **^**Weyl 1935- ^
^{a}^{b}^{c}Lederman & Hill 2004, p. 73. - ^
^{a}^{b}Dick 1981, p. 128 **^**Dick 1981, pp. 7–8.**^**Dick 1981, pp. 9–10.**^**Dick 1981, pp. 10–11.**^**Dick 1981, p. 15.**^**Dick 1981, pp. 15, 19–20.**^**Dick 1981, pp. 25, 45.**^**Kimberling 1981, p. 5.- ^
^{a}^{b}Dick 1981, pp. 11–12. **^**Dick 1981, pp. 15–16.**^**Kimberling 1981, p. 10.**^**Kimberling 1981, pp. 8–10.**^**Lederman & Hill 2004, p. 71.**^**Dick 1981, p. 14.**^**Rowe 2021, p. 18.**^**Dick 1981, pp. 14–15.- ^
^{a}^{b}Noether 1908. **^**Dick 1981, pp. 16–18.**^**Merzbach 1983, p. 164.- ^
^{a}^{b}Kimberling 1981, pp. 10–11. **^**Dick 1981, pp. 13–17.**^**Dick 1981, pp. 18, 24.**^**Kosmann-Schwarzbach 2011, p. 44.**^**Dick 1981, p. 23.**^**Rowe 2021, p. 22.- ^
^{a}^{b}^{c}^{d}^{e}^{f}Weyl 1935. **^**Dick 1981, pp. 23–24.- ^
^{a}^{b}Kimberling 1981, pp. 11–12. **^**Dick 1981, pp. 18–24.**^**Rowe 2021, pp. 29–35.- ^
^{a}^{b}Rowe & Koreuber 2020, p. 27. - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}O'Connor, John J.; Robertson, Edmund F. (November 2014), "Emmy Noether's doctoral students",*MacTutor History of Mathematics Archive*, University of St Andrews **^**Falckenberg 1912.**^**Seidelmann 1917.**^**Dick 1981, p. 16.- ^
^{a}^{b}Kimberling 1981, p. 14. - ^
^{a}^{b}Lederman & Hill 2004, p. 72. **^**Rowe & Koreuber 2020, pp. 75–76.- ^
^{a}^{b}Dick 1981, p. 32. **^**Dick 1981, pp. 24–26.- ^
^{a}^{b}Byers 2006, pp. 91–92. **^**Byers 2006, p. 86.**^**Noether 1918c, p. 235.**^**Rowe & Koreuber 2020, p. 3.**^**Byers 1996, p. 2.**^**Dick 1981, pp. 32–24.**^**Kosmann-Schwarzbach 2011, p. 49.**^**Dick 1981, pp. 36–37.- ^
^{a}^{b}Dick 1981, p. 188. **^**Kimberling 1981, pp. 14–18.**^**Dick 1981, pp. 33–34.- ^
^{a}^{b}Noether 1921. - ^
^{a}^{b}Kimberling 1981, p. 18. **^**Dick 1981, pp. 44–45.- ^
^{a}^{b}^{c}^{d}^{e}van der Waerden 1935. - ^
^{a}^{b}^{c}^{d}O'Connor, John J.; Robertson, Edmund F., "Emmy Amalie Noether",*MacTutor History of Mathematics Archive*, University of St Andrews - ^
^{a}^{b}Lederman & Hill 2004, p. 74. **^**Dick 1981, pp. 57–58.**^**Kimberling 1981, p. 19.**^**Kimberling 1981, p. 24.- ^
^{a}^{b}Kimberling 1981, pp. 24–25. **^**Dick 1981, pp. 61–63.- ^
^{a}^{b}Segal 2003, p. 128. **^**Dick 1981, pp. 51–53. See p. 51: "... Grete Hermann who took her examinations in February 1925 with E. Noether and E. Landau, See also pp. 52–53: "In 1929 Werner Weber obtained a doctor's degree ... The reviewers were E. Landau and E. Noether." Also on p. 53: "He was followed two weeks later by Jakob Levitzki ... who also was examined by Noether and Landau.- ^
^{a}^{b}^{c}^{d}Dick 1981, p. 51. **^**Hermann 1926.**^**Rowe 2021, p. 99.**^**Grell 1927.**^**Hölzer 1927.**^**Weber 1930.**^**Levitzki 1931.**^**Segal 2003, pp. 128–129.- ^
^{a}^{b}Dick 1981, p. 53. - ^
^{a}^{b}^{c}Kimberling 1981, p. 29. **^**Deuring 1932.- ^
^{a}^{b}Kimberling 1981, p. 40. **^**Dick 1981, p. 54.**^**Dick 1981, pp. 53–54.**^**Fitting 1933.- ^
^{a}^{b}^{c}^{d}Kimberling 1981, p. 41. - ^
^{a}^{b}Dick 1981, p. 55. **^**Witt 1935.**^**Tsen 1933.**^**"十月份甯屬要聞" [Main news of Ningshu in October].*新寧遠月刊 Xin Ningyuan Yuekang [New Ningyuan Monthly]*(in Chinese). Vol. 1, no. 3. Xichang, Xikang. 25 November 1940. p. 51.一日 國立西康技藝專科學校教授曾烱之博士在西康衞生院病逝。 [1st: Dr. Chiungtze Tsen, professor at National Xikang Institute of Technology, died from illness in Xikang Health Center.]

**^**Schilling 1935.**^**Dörnte 1929.**^**Wichmann 1936.**^**Rowe 2021, p. 200.**^**Rowe & Koreuber 2020, p. 32.**^**Dick 1981, pp. 56–57.**^**Rowe 2021, p. x.- ^
^{a}^{b}Dick 1981, p. 57. **^**Dick 1981, pp. 37–49.**^**Mac Lane 1981, p. 71.**^**Dick 1981, p. 76.**^**Dick 1981, pp. 46–48.**^**Taussky 1981, p. 80.**^**Dick 1981, pp. 40–41.**^**Rowe & Koreuber 2020, p. 21, 122.**^**Dick 1981, pp. 37–38.**^**Mac Lane 1981, p. 77.**^**Dick 1981, p. 41.**^**Rowe & Koreuber 2020, pp. 36, 99.**^**Dick 1981, p. 38.**^**Scharlau, Winfried,*Emmy Noether's Contributions to the Theory of Algebras*in Teicher 1999, p. 49.**^**Dick 1981, pp. 63–64.**^**Kimberling 1981, p. 26.**^**Alexandrov 1981, pp. 108–110.- ^
^{a}^{b}Alexandrov 1981, pp. 106–109. **^**McLarty 2005.- ^
^{a}^{b}Dick 1981, pp. 72–73. - ^
^{a}^{b}^{c}Kimberling 1981, pp. 26–27. **^**Hasse 1933, p. 731.**^**Dick 1981, pp. 74–75.- ^
^{a}^{b}^{c}Dick 1981, pp. 75–76. - ^
^{a}^{b}^{c}Kimberling 1981, pp. 28–29. **^**Dick 1981, pp. 78–79.**^**Kimberling 1981, pp. 30–31.**^**Kimberling 1981, pp. 32–33.**^**Dick 1981, p. 80.**^**Rowe 2021, p. 222.**^**Rowe 2021, pp. 223.**^**Dick 1981, pp. 80–81.**^**Dick 1981, pp. 85–86.- ^
^{a}^{b}Rowe 2021, p. 251. **^**Stauffer 1936.**^**"Emmy Noether at the Institute for Advanced Study".*StoryMaps*. ArcGIS. 7 December 2019. Archived from the original on 16 April 2024. Retrieved 28 August 2020.**^**Dick 1981, pp. 81–82.**^**Dick 1981, p. 81.**^**Dick 1981, p. 83.- ^
^{a}^{b}Dick 1981, pp. 82–83. **^**O'Connor, John J.; Robertson, Edmund F., "Fritz Alexander Ernst Noether",*MacTutor History of Mathematics Archive*, University of St Andrews.**^**Dick 1981, p. 82.**^**Kimberling 1981, p. 34.- ^
^{a}^{b}Kimberling 1981, pp. 37–38. **^**Rowe 2021, p. 252.**^**Rowe 2021, pp. 252, 257.**^**Kimberling 1981, p. 39.- ^
^{a}^{b}Chodos, Alan, ed. (March 2013). "This Month in Physics History: March 23, 1882: Birth of Emmy Noether".*APSNews*.**22**(3). American Physical Society. Archived from the original on 14 July 2024. Retrieved 28 August 2020. **^**Gilmer 1981, p. 131.**^**Kimberling 1981, pp. 10–23.- ^
^{a}^{b}Rowe & Koreuber 2020, pp. 27–30. See p. 27: "In 1921, Noether published her famous paper ... [which] dealt with rings whose ideals satisfy the ascending chain condition". On the same page, Olaf Neumann is quoted as calling this paper "a work of genius which showed the amazing consequences of the ascending chain condition". See also 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. **^**Gauss, Carl F. (1832). "Theoria residuorum biquadraticorum – Commentatio secunda".*Comm. Soc. Reg. Sci. Göttingen*(in Latin).**7**: 1–34. Reprinted in*Werke*[*Complete Works of C.F. Gauss*]. Hildesheim: Georg Olms Verlag. 1973. pp. 93–148.**^**G.E. Noether 1987, p. 168.**^**Lang 2005, p. 16, II.§1.**^**Stewart 2015, pp. 18–19.**^**Stewart 2015, p. 182.**^**Stewart 2015, p. 183.**^**Gowers et al. 2008, p. 284.**^**Gowers et al. 2008, pp. 699–700.**^**Osofsky 1994.**^**Zee 2016, pp. 89–92.**^**Peres 1993, pp. 215–229.- ^
^{a}^{b}Zee 2016, p. 180. **^**Gowers et al. 2008, p. 285.**^**Givant & Halmos 2009, pp. 14–15.**^**Dick 1981, p. 101.**^**Gowers et al. 2008, p. 801.**^**Noether 1914, p. 11.**^**Gordan 1870.**^**Weyl 1944, pp. 618–621.**^**Hilbert 1890, p. 531.**^**Hilbert 1890, p. 532.**^**Dick 1981, pp. 16–18, 155–156.**^**Dick 1981, p. 120.**^**Stewart 2015, pp. 108–111**^**Stewart 2015, pp. 22–23.**^**Stewart 2015, pp. 23, 39.**^**Stewart 2015, pp. 39, 129.**^**Stewart 2015, pp. 44, 129, 148.**^**Stewart 2015, pp. 129–130**^**Stewart 2015, pp. 112–114**^**Stewart 2015, pp. 114–116, 151–153**^**Noether 1918.**^**Noether 1913.**^**Swan 1969, p. 148.**^**Malle & Matzat 1999.**^**Gowers et al. 2008, p. 800.**^**Noether 1918b**^**Kosmann-Schwarzbach 2011, p. 25**^**Lynch, Peter (18 June 2015). "Emmy Noether's beautiful theorem".*ThatsMaths*. Archived from the original on 9 December 2023. Retrieved 28 August 2020.**^**Kimberling 1981, p. 13**^**Lederman & Hill 2004, pp. 97–116.**^**Taylor 2005, pp. 268–272.**^**Kosmann-Schwarzbach 2011, pp. 26, 101–102**^**Goodearl & Warfield Jr. 2004, p. 1.**^**Atiyah & MacDonald 1994, p. 74.- ^
^{a}^{b}^{c}Gray 2018, p. 294. **^**Goodearl & Warfield Jr. 2004, pp. 1–3.- ^
^{a}^{b}Hartshorne 1977, p. 5. **^**Atiyah & MacDonald 1994, p. 79.- ^
^{a}^{b}Gilmer 1981, p. 133 **^**Osofsky 1994**^**Noether 1927.**^**Noether 1923.**^**Noether 1923b.**^**Noether 1924.**^**Cox, Little & O'Shea 2015, p. 121.**^**Noether 1915.**^**Fleischmann 2000, p. 24.**^**Fleischmann 2000, p. 25.**^**Fogarty 2001, p. 5.**^**Noether 1926.**^**Haboush 1975.**^**Hubbard & West 1991, p. 204.**^**Hilton 1988, p. 284.**^**Dick 1981, p. 173.- ^
^{a}^{b}Dick 1981, p. 174. **^**Hirzebruch, Friedrich,*Emmy Noether and Topology*in Teicher 1999, pp. 57–61**^**Hopf 1928.**^**Dick 1981, pp. 174–175.**^**Noether 1926b.**^**Hirzebruch, Friedrich,*Emmy Noether and Topology*in Teicher 1999, p. 63**^**Hirzebruch, Friedrich,*Emmy Noether and Topology*in Teicher 1999, pp. 61–63**^**Noether 1929.- ^
^{a}^{b}Rowe 2021, p. 127. **^**van der Waerden 1985, p. 244.**^**Lam 1981, pp. 152–53.**^**Brauer, Hasse & Noether 1932.**^**Noether 1933.**^**Brauer & Noether 1927.**^**Roquette 2005, p. 6.**^**Plumer, Brad (23 March 2016). "Emmy Noether revolutionized mathematics — and still faced sexism all her life".*Vox*. Archived from the original on 7 September 2024. Retrieved 2 November 2024.**^**James 2002, p. 321.**^**Dick 1981, p. 154.**^**Dick 1981, p. 152.**^**Noether 1987, p. 167.**^**Dick 1981, p. 100.**^**Schmadel 2003, p. 270.

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- Emmy Noether at the Mathematics Genealogy Project
- Emmy Noether on
*In Our Time*at the BBC - Noether's application for admission to the University of Erlangen and three of her curriculum vitae from the website of historian Cordula Tollmien
- Letter by Noether to Dr. Park, Bryn Mawr College President – Bryn Mawr College Library Special Collections
- Photograph of Noether taken by Hanna Kunsch – Bryn Mawr College Library Special Collections
- Photographs of Noether – Oberwolfach Photo Collection of the Mathematisches Forschungsinstitut Oberwolfach
- Photographs of Noether's colleagues and acquaintances from the website of Clark Kimberling