Editing Emmy Noether (section)

== Biography ==
=== Early life ===
[[File:Erlangen 1916.jpg|thumb|Noether grew up in the Bavarian city of [[Erlangen]], depicted here in a 1916 postcard.|alt=1916 postcard depicting Universitätstraße in Erlangen]]
Amalie Emmy Noether was born on 23 March 1882 in [[Erlangen]], Bavaria.{{sfn|Dick|1981|p=4}} She was the first of four children of mathematician [[Max Noether]] and Ida Amalia Kaufmann, both from wealthy Jewish merchant families.{{sfn|Dick|1981|pp=7–8}} 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.{{refn|group=lower-alpha|name=Rufname}}

In her youth, Noether did not stand out academically, although she was known for being clever and friendly. She was [[myopia|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.{{sfn|Dick|1981|pp=9–10}} 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.{{Sfn|Dick|1981|pp=10–11}}

[[File:NoetherFamily MFO3120.jpg|thumb|left|Emmy Noether with her brothers Alfred, [[Fritz Noether|Fritz]], and Robert, before 1918|alt=Family portrait of Alfred, Emmy, Fritz and Robert Noether]]
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.{{Sfn|Dick|1981|p=15}} [[Fritz Noether]] was born in 1884, studied in [[Munich]] and made contributions to [[applied mathematics]].{{Sfn|Dick|1981|pp=15, 19–20}} He was likely executed in the Soviet Union in 1941.<ref>{{MacTutor|id=Noether_Fritz |title=Fritz Alexander Ernst Noether}}.</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.{{Sfn|Dick|1981|pp=25, 45}}{{Sfn|Kimberling|1981|p=5}}

=== Education ===
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]],{{Sfn |Dick|1981|pp=11–12}} at which her father was a professor.{{Sfn |Dick|1981|pp=15–16}}

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".{{Sfn |Kimberling|1981|p=10}} One of just two women in a university of 986&nbsp;students, Noether was allowed only to [[academic audit|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&nbsp;July 1903, she passed the graduation exam at a ''[[Realgymnasium]]'' in [[Nuremberg]].{{Sfn |Dick|1981|pp=11–12}}{{Sfn |Kimberling|1981|pp=8–10}}{{Sfn |Lederman|Hill|2004|p= 71}}

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]].{{Sfn |Dick|1981|p=14}}

[[File:Paul Albert Gordan.jpg|thumb|[[Paul Gordan]] supervised Noether's doctoral dissertation on [[invariant (mathematics)|invariants]] of biquadratic forms.]]
In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded.{{sfn|Rowe|2021|p=18}} 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.{{Sfn |Dick|1981|pp=14–15}} 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''),{{sfn|Noether|1908}} in 1907, graduating ''summa cum laude'' later that year.{{Sfn |Dick|1981|pp=16–18}} 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.{{Sfn|Merzbach|1983|p=164}}{{Sfn|Kimberling|1981|pp=10–11}} 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.{{Sfn|Kimberling|1981|pp=10–11}}{{Sfn |Dick|1981|pp=13–17}}

=== University of Erlangen–Nuremberg ===
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.{{Sfn |Dick|1981|pp=18, 24}} She joined the [[Circolo Matematico di Palermo]] in 1908 and the [[German Mathematical Society|Deutsche Mathematiker-Vereinigung]] in 1909.{{sfn|Dick|1981|p=18}} In 1910 and 1911, she published an extension of her thesis work from three variables to ''n'' variables.{{Sfn |Kosmann-Schwarzbach|2011|p=44}}

[[File:Emmy noether postcard 1915.jpg|thumb|upright=1.2|Noether sometimes used postcards to discuss abstract algebra with her colleague, [[Ernst Sigismund Fischer|Ernst Fischer]]. This card is postmarked 10&nbsp;April 1915.]]
Gordan retired in 1910,{{Sfn |Dick|1981|p=23}} and Noether taught under his successors, [[Erhard Schmidt]] and [[Ernst Sigismund Fischer|Ernst Fischer]], who took over from the former in 1911.{{Sfn |Rowe|2021|p=22}} 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]].{{Sfn |Weyl|1935}}{{Sfn |Dick|1981|pp=23–24}} 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.{{Sfn|Kimberling|1981|pp=11–12}}{{Sfn|Dick|1981|pp=18–24}}

From 1913 to 1916, Noether published several papers extending and applying Hilbert's methods to mathematical objects such as [[field (mathematics)|fields]] of [[rational function]]s and the [[invariant theory|invariants]] of [[finite group]]s.{{Sfn |Rowe|2021|pp=29–35}} This phase marked Noether's first exposure to [[abstract algebra]], the field to which she would make groundbreaking contributions.{{sfn|Rowe|Koreuber|2020|p=27}}

In Erlangen, Noether advised two doctoral students:<ref name="MacTutorStudents">{{MacTutor|class=Extras|id=Noether_students|title=Emmy Noether's doctoral students |date=November 2014}}</ref> Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916.{{sfn|Falckenberg|1912}}{{sfn|Seidelmann|1917}} 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]]{{sfn|Dick|1981|p=16}} while Seidelmann became a professor in [[Munich]].<ref name="MacTutorStudents"/>

=== University of Göttingen ===
==== Habilitation and Noether's theorem ====
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 [[Philology|philologists]] and [[historian]]s among the philosophical faculty, who insisted that women should not become ''[[privatdozent]]en''. 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?"{{Sfn|Kimberling|1981|p=14}}{{Sfn|Lederman|Hill|2004|p=72}} 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."{{Sfn|Weyl|1935}}{{Sfn|Kimberling|1981|p=14}}{{sfn|Rowe|Koreuber|2020|pp=75–76}}{{Sfn|Dick|1981|p=32}} According to [[Pavel Alexandrov]]'s recollection, faculty members' opposition to Noether was based not just in sexism, but also in their objections to her [[Socialist democracy|social-democratic]] political beliefs and Jewish ancestry.{{sfn|Dick|1981|p=32}}

[[File:Hilbert.jpg|thumb|left|upright|[[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 (German Empire)|German Army]] to serve in [[World War I]]. She returned to Erlangen for several weeks, mostly to care for her aging father.{{Sfn|Dick|1981|pp=24–26}}

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".{{Sfn |Byers|2006|pp=91–92}}

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 (physics)|conservation law]] is associated with any differentiable [[symmetry in physics|symmetry of a physical system]].{{Sfn|Lederman|Hill|2004|p=72}}{{sfn|Byers|2006|p=86}} 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.{{Sfn|Noether|1918c|p=235}}{{sfn|Rowe|Koreuber|2020|p=3}} Noether presumably did not present it herself because she was not a member of the society.{{Sfn|Byers|1996|p=2}} 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]]".{{Sfn|Lederman|Hill|2004|p=73}}

[[File:Mathematik Göttingen.jpg|thumb|210px|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.{{Sfn |Dick|1981|pp=32–24}} Noether became a ''privatdozent'',{{Sfn |Kosmann-Schwarzbach|2011|p=49}} and she delivered that fall semester the first lectures listed under her own name.{{Sfn |Dick|1981|pp=36–37}} She was still not paid for her work.{{Sfn |Byers|2006|pp=91–92}}

Three years later, she received a letter from {{ill|Otto Boelitz|de}}, the [[Prussia]]n 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).{{Sfn|Dick|1981|p=188}} This was an unpaid "extraordinary" [[professor]]ship, 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.{{Sfn|Kimberling|1981|pp=14–18}}{{Sfn|Dick|1981|pp=33–34}}

====Work in abstract algebra====
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<blockquote>The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her&nbsp;— in published papers, in lectures, and in personal influence on her contemporaries.{{sfn|Noether|1983}}</blockquote>

Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published a paper about the [[ideal theory|theory of ideals]] in which they defined [[Ideal (ring theory)|left and right ideals]] in a [[ring (mathematics)|ring]].{{sfn|Rowe|Koreuber|2020|p=27}}

The following year she published the paper ''Idealtheorie in Ringbereichen'',{{Sfn | Noether | 1921}} analyzing [[ascending chain condition]]s with regards to (mathematical) [[Ideal (ring theory)|ideals]], in which she proved the [[Lasker–Noether theorem]] in its full generality. Noted algebraist [[Irving Kaplansky]] called this work "revolutionary".{{Sfn |Kimberling|1981|p=18}} The publication gave rise to the term ''[[Noetherian]]'' for objects which satisfy the ascending chain condition.{{Sfn|Kimberling|1981|p=18}}{{Sfn|Dick|1981|pp=44–45}}

[[File:ETH-BIB-Waerden, Bartel Leendert van der (1903-1996)-Portr 12109.tif|thumb|230x230px|[[Bartel Leendert van der Waerden|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".{{Sfn|van der Waerden|1935}} 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 [[Emil Artin|E. Artin]] and E. Noether".{{Sfn|Lederman|Hill|2004|p=74}}{{Sfn|Dick|1981|pp=57–58}}{{Sfn|Kimberling|1981|p=19}} Beginning in 1927, Noether worked closely with [[Emil Artin]], [[Richard Brauer]] and [[Helmut Hasse]] on [[noncommutative algebra]]s.{{sfn |Weyl| 1935}}<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.{{Sfn|Kimberling|1981|p=24}} Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends.{{Sfn|Kimberling|1981|pp=24–25}} He dubbed her ''der Noether'', using ''der'' as an epithet rather than as the masculine German article.{{efn|The nickname was not always used in a well-meaning manner.{{sfn|Rowe|Koreuber|2020|p=14}} In Noether's obituary, Hermann stated that <blockquote>The power of your genius seemed to transcend the bounds of your sex, which is why we in Göttingen, in awed mockery, often spoke of you in the masculine form as "der Noether."{{sfn|Weyl|1935}}{{sfn|Rowe|Koreuber|2020|p=214}}</blockquote>}}{{Sfn|Kimberling|1981|pp=24–25}} 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]].{{Sfn|Kimberling|1981|pp=24–25}}{{Sfn|Dick|1981|pp=61–63}}

====Graduate students====
[[File:EmmyNoether MFO3096.jpg|thumb|left|Noether c.&nbsp;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.{{sfn|Segal|2003|p=128}}{{sfn|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}} Her first was [[Grete Hermann]], who defended her dissertation in February&nbsp;1925.{{Sfn|Dick|1981|p=51}} Although she is best remembered for her work on the foundations of [[quantum mechanics]], her dissertation was considered an important contribution to [[ideal theory]].{{sfn|Hermann|1926}}{{sfn|Rowe|2021|p=99}} Hermann later spoke reverently of her "dissertation-mother".{{Sfn|Dick|1981|p=51}}

Around the same time, Heinrich Grell and Rudolf Hölzer wrote their dissertations under Noether, though the latter died of [[tuberculosis]] shortly before his defense.{{Sfn|Dick|1981|p=51}}{{sfn|Grell|1927}}{{sfn|Hölzer|1927}} 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"/>{{Sfn|Dick|1981|p=51}}

Noether then supervised [[Werner Weber (mathematician)|Werner Weber]]{{sfn|Weber|1930}} and [[Jakob Levitzki]],{{sfn|Levitzki|1931}} who both defended their theses in 1929.{{sfn|Segal|2003|pp=128–129}}{{sfn|Dick|1981|p=53}} Weber, who was considered only a modest mathematician,{{sfn|Segal|2003|p=128}} would later take part in driving Jewish mathematicians out of Göttingen.{{sfn|Kimberling|1981|p=29}} 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]].{{sfn|Dick|1981|p=53}}

Other <em>Noether Boys</em> 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.{{sfn|Deuring|1932}}{{sfn|Kimberling|1981|p=40}} He worked in Hamburg, Marden and Göttingen{{efn|When Noether was forced to leave Germany in 1933, she wished for the university to appoint Deuring as her successor,{{sfn|Dick|1981|p=54}} but he only started teaching there in 1950.{{sfn|Kimberling|1981|p=40}}}} and is known for his contributions to [[arithmetic geometry]].{{sfn|Dick|1981|pp=53–54}} Fitting graduated in 1931 with a thesis on abelian groups{{sfn|Fitting|1933}} and is remembered for his work in [[group theory]], particularly [[Fitting's theorem]] and the [[Fitting lemma]].{{sfn|Kimberling|1981|p=41}} He died at the age of 31 from a bone disease.{{sfn|Dick|1981|p=55}}

Witt was initially supervised by Noether, but her position was revoked in April 1933 and he was assigned to [[Gustav Herglotz]] instead.{{sfn|Dick|1981|p=55}} He received his PhD in July 1933 with a thesis on the [[Riemann-Roch theorem]] and [[zeta-function]]s,{{sfn|Witt|1935}} and went on to make several contributions that [[List of things named after Ernst Witt|now bear his name]].{{sfn|Kimberling|1981|p=41}} Tsen, best remembered for proving [[Tsen's theorem]], received his doctorate in December of the same year.{{sfn|Tsen|1933}} He returned to [[China]] in 1935 and started teaching at [[National Chekiang University]],{{sfn|Kimberling|1981|p=41}} but died only five years later.{{efn|Accounts of Tsen's date of death vary: {{harvtxt|Kimberling|1981|p=41}} states that he died "some time in 1939 or 40" and {{harvtxt|Ding|Kang|Tan|1999}} state that he died in November 1940, but a local newspaper recorded his date of death as 1 October 1940.<ref>{{cite news|title=十月份甯屬要聞|trans-title=Main news of Ningshu in October|newspaper=新寧遠月刊 Xin Ningyuan Yuekang [New Ningyuan Monthly]|volume=1|issue=3|date=25 November 1940|place=[[Xichang]], [[Xikang]]|language=Chinese|page=51|quote=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.]|url=https://upload.wikimedia.org/wikipedia/commons/6/67/Ningshu_News_October_1940_Zeng_Jiongzhi_died.jpg}}</ref>}} 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]].{{sfn|Kimberling|1981|p=41}}{{sfn|Schilling|1935}} 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,{{sfn|Dörnte|1929}} Werner Vorbeck, who did so in 1935 with a thesis on [[splitting field]]s,<ref name="MacTutorStudents"/> and Wolfgang Wichmann, who did so 1936 with a thesis on [[p-adic number|p-adic theory]].{{sfn|Wichmann|1936}} 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{{sfn|Rowe|2021|p=200}} and died as a soldier on the [[Eastern Front (World War II)|Eastern Front]] during [[World War II]].<ref name="MacTutorStudents"/>

====Noether school====
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,{{sfn|Rowe|Koreuber|2020|p=32}} a group often referred to as the <em>Noether school</em>.{{Sfn|Dick|1981|pp=56–57}}{{sfn|Rowe|2021|p=x}} An example of this is her close work with [[Wolfgang Krull]], who greatly advanced [[commutative algebra]] with his [[Krull's principal ideal theorem|''Hauptidealsatz'']] and his [[Krull dimension|dimension theory]] for commutative rings.{{Sfn|Dick|1981|p=57}} Another is [[Gottfried Köthe]], who contributed to the development of the theory of [[hypercomplex number|hypercomplex quantities]] using Noether and Krull's methods.{{Sfn|Dick|1981|p=57}}

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.{{Sfn|Dick|1981|pp=37–49}} In Noether's obituary, Van der Waerden described her as<blockquote>Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all.{{Sfn|van der Waerden|1935}}</blockquote>

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.{{Sfn |Mac Lane|1981|p=71}} Later, after [[Nazi Germany]] dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts.{{Sfn |Dick|1981|p= 76}}

====Influential lectures====
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]].{{Sfn|Dick|1981|pp=46–48}}

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".{{Sfn|Taussky|1981|p=80}} 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.{{Sfn|Dick|1981|pp=40–41}}

Noether did not follow a lesson plan for her lectures.{{Sfn|van der Waerden|1935}} She spoke quickly and her lectures were considered difficult to follow by many, including [[Carl Ludwig Siegel]] and [[Paul Dubreil]].{{sfn|Rowe|Koreuber|2020|p=21, 122}}{{Sfn|Dick|1981|pp=37–38}} Students who disliked her style often felt alienated.{{sfn|Mac Lane|1981|p=77}} "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."{{sfn|Dick|1981|p=41}}

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.{{Sfn|van der Waerden|1935}} Noether transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her.{{sfn|Rowe|Koreuber|2020|pp=36, 99}}{{sfn|Dick|1981|p=38}}

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">{{MacTutor|id=Noether_Emmy |title=Emmy Amalie Noether}}</ref>{{Sfn|Lederman|Hill|2004|p=74}} Noether was recorded as having given at least five semester-long courses at Göttingen:<ref name="scharlau_49">{{citation |last=Scharlau |first=Winfried |author-link=Winfried Scharlau |title=Emmy Noether's Contributions to the Theory of Algebras}} in {{Harvnb|Teicher|1999|p=49}}.</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 University===
[[File:Paul S Alexandroff 2.jpg|thumb|140px|[[Pavel Alexandrov]]]]
In the winter of 1928–1929, Noether accepted an invitation to [[Moscow State University]], where she continued working with [[Pavel Alexandrov|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]].{{Sfn|Dick|1981|pp=63–64}}{{Sfn|Kimberling|1981|p=26}}{{Sfn|Alexandrov|1981|pp=108–110}}

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 Union|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)|pension lodging]] building, after student leaders complained of living with "a Marxist-leaning Jewess".{{Sfn|Alexandrov|1981|pp=106–109}} [[Hermann Weyl]] recalled that "During the wild times after the [[German Revolution of 1918–1919|Revolution of 1918]]," Noether "sided more or less with the [[Social Democratic Party of Germany|Social Democrats]]".{{sfn|Weyl|1935}} She was from 1919 through 1922 a member of the [[Independent Social Democratic Party of Germany|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."{{sfn|McLarty|2005}}

[[File:Moscow 05-2012 Mokhovaya 05.jpg|thumb|left|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 [[Narkompros|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.{{Sfn |Alexandrov|1981|pp=106–109}}

===Recognition===
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 {{Reichsmark|500|link=yes}} 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 Academy of Sciences|Göttingen ''Gesellschaft der Wissenschaften'']] (academy of sciences) and was never promoted to the position of ''[[Ordentlicher Professor]]''{{Sfn|Dick|1981|pp=72–73}}{{Sfn |Kimberling|1981|pp=26–27}} (full professor).{{Sfn|Dick|1981|p=188}}

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 [[quadratic reciprocity|reciprocity law]].{{Sfn|Hasse|1933|p=731}} This pleased her immensely. He also sent her a mathematical riddle, which he called the "m<sub>μν</sub>-riddle of syllables". She solved it immediately, but the riddle has been lost.{{Sfn|Dick|1981|pp=72–73}}{{Sfn|Kimberling|1981|pp=26–27}}

[[File:Internationaler Mathematikerkongress Zürich 1932 - ETH BIB Portr 10680-FL (Johannes Meiner).jpg|thumb|Noether visited [[Zürich]] in 1932 to deliver a [[list of International Congresses of Mathematicians Plenary and Invited Speakers|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&nbsp;people, including Noether's colleagues [[Hermann Weyl]], [[Edmund Landau]], and [[Wolfgang Krull]]. There were 420&nbsp;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&nbsp;congress is sometimes described as the high point of her career.{{sfn|Kimberling|1981|pp=26–27}}{{sfn|Dick|1981|pp=74–75}}

===Expulsion from Göttingen by Nazi Germany===
When [[Adolf Hitler]] became the [[Chancellor of Germany (German Reich)|German ''Reichskanzler'']] in January&nbsp;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 (mathematician)|Werner Weber]]. [[Antisemitism|Antisemitic]] attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want [[Deutsche Mathematik|Aryan mathematics]] and not Jewish mathematics."{{sfn|Kimberling|1981|p=29}}

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 [[Frontkämpferprivileg|serving in World War&nbsp;I]]. In April&nbsp;1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph&nbsp;3 of the Civil Service Code of 7&nbsp;April 1933, I hereby withdraw from you the right to teach at the University of Göttingen."{{Sfn|Dick|1981|pp=75–76}}{{sfn |Kimberling|1981|pp=28–29}} Several of Noether's colleagues, including [[Max Born]] and [[Richard Courant]], also had their positions revoked.{{sfn|Dick|1981|pp=75–76}}{{sfn|Kimberling|1981|pp=28–29}}

Noether accepted the decision calmly, providing support for others during this difficult time. [[Hermann Weyl]] later wrote that "Emmy Noether{{snd}}her courage, her frankness, her unconcern about her own fate, her conciliatory spirit{{snd}}was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace."{{sfn|Kimberling|1981|p=29}} 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.{{sfn|Dick|1981|pp=75–76}}{{sfn|Kimberling|1981|pp=28–29}}

===Refuge at Bryn Mawr and Princeton===
[[File:Entrance Bryn Mawr.JPG|thumb|right|[[Bryn Mawr College]] provided a welcoming home for Noether during the last two&nbsp;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, New Jersey|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.{{Sfn|Dick|1981|pp=78–79}}{{Sfn|Kimberling|1981|pp=30–31}}

At Bryn Mawr, Noether met and befriended [[Anna Johnson Pell Wheeler|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.&nbsp;Noether in action!"{{Sfn|Kimberling|1981|pp=32–33}}{{Sfn|Dick|1981|p=80}}

During her time at Bryn Mawr, Noether formed a group, sometimes called the <em>Noether girls,</em>{{sfn|Rowe|2021|p=222}} 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).{{sfn|Rowe|2021|pp=223}} They enthusiastically worked through [[van der Waerden]]'s ''Moderne Algebra&nbsp;I'' and parts of [[Erich Hecke]]'s ''Theorie der algebraischen Zahlen'' (''Theory of algebraic numbers'').{{sfn|Dick|1981|pp=80–81}} Stauffer was Noether's only doctoral student in the United States, but Noether died shortly before she graduated.{{sfn|Dick|1981|pp=85–86}} She took her examination with [[Richard Brauer]] and received her degree in June 1935,{{sfn|Rowe|2021|p=251}} with a thesis concerning separable [[normal extension]]s.{{sfn|Stauffer|1936}} After her doctorate, Stauffer worked as a teacher for a short period and as a statistician for over 30 years.<ref name="MacTutorStudents"/>{{sfn|Rowe|2021|p=251}}

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of [[Abraham Flexner]] and [[Oswald Veblen]].<ref>{{cite web |title=Emmy Noether at the Institute for Advanced Study |url=https://storymaps.arcgis.com/stories/e7329da167ae4fd690da903f2610432d |website=StoryMaps |date=7 December 2019 |publisher=[[ArcGIS]] |access-date=28 August 2020 |archive-url=https://web.archive.org/web/20240416231133/https://storymaps.arcgis.com/stories/e7329da167ae4fd690da903f2610432d |archive-date=16 April 2024 |url-status=live}}</ref> She also worked with [[Abraham Adrian Albert|Abraham Albert]] and [[Harry Vandiver]].{{Sfn|Dick|1981|pp=81–82}} However, she remarked about [[Princeton University]] that she was not welcome at "the men's university, where nothing female is admitted".{{Sfn|Dick|1981|p=81}}

Her time in the United States was pleasant, as she was surrounded by supportive colleagues and absorbed in her favorite subjects.{{Sfn|Dick|1981|p=83}} In the summer of 1934, she briefly returned to Germany to see Emil Artin and her brother [[Fritz Noether|Fritz]].{{Sfn|Dick|1981|pp=82–83}} 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.{{Sfn|Dick|1981|pp=82–83}}

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.{{Sfn|Dick|1981|p=82}}{{Sfn|Kimberling|1981|p=34}}

===Death===
In April 1935, doctors discovered a [[Neoplasm|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]]".{{Sfn |Kimberling| 1981| pp= 37–38}} 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 {{convert|109|°F|°C|sigfig=3}}, 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.{{Sfn|Kimberling|1981|pp=37–38}}

[[File:Bryn Mawr College Cloisters.JPG|thumb|250px|right|Noether's ashes were placed under the cloistered walkway of Bryn Mawr's [[Bryn Mawr College#Old Library (previously M. Carey Thomas Library and College Hall)|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.{{sfn|Rowe|2021|p=252}} Hermann Weyl and Richard Brauer both traveled from Princeton and delivered eulogies.{{sfn|Rowe|2021|pp=252, 257}} 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 [[Bryn Mawr College#Old Library (previously M. Carey Thomas Library and College Hall)|M. Carey Thomas Library]] at Bryn Mawr.{{Sfn|Kimberling|1981|p= 39}}<ref name="APSNews">{{cite journal |journal=APSNews| title=This Month in Physics History: March 23, 1882: Birth of Emmy Noether |url=https://www.aps.org/publications/apsnews/201303/physicshistory.cfm |editor-last=Chodos |editor-first=Alan |publisher=[[American Physical Society]] |access-date=28 August 2020 |language=en |date=March 2013 |number=3 |volume=22 |archive-url=https://web.archive.org/web/20240714164817/https://www.aps.org/archives/publications/apsnews/201303/physicshistory.cfm |archive-date=14 July 2024 |url-status=live}}</ref>