Editing Emmy Noether (section)

=== 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'']