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{{Infobox Scientist
| box_width         = 250px
| name              = Emmy Noether
| image             = Noether.jpg
| caption           = Amalie Emmy Noether
| birth_date        = {{birth-date|23 March 1882}} 
| birth_place       = [[Erlangen]], [[Bavaria]], Germany
| death_date = {{death-date|14 April 1935 }} (aged 53)
| death_place       = {{city-state|Bryn Mawr|Pennsylvania}}, [[United States|USA]]
| residence         = 
| citizenship       = Germany (1882–1933)<br />United States (1933–35)
| ethnicity         = 
| fields            = [[Mathematics]] and [[Physics]]
| workplaces        = [[University of Göttingen]]<br />[[Bryn Mawr College]]
| alma_mater        = [[University of Erlangen]]
| doctoral_advisor  = [[Paul Gordan]]
| academic_advisors = 
| doctoral_students = [[Grete Hermann]]<br />[[Max Deuring]]<br />[[Hans Fitting]]<br />[[Zeng Jiongzhi]]
| known_for         = [[Abstract algebra]]<br />[[Theoretical physics]]
| awards            = 
| religion          = 
| signature         = 
| footnotes         = <!--For any footnotes needed to clarify entries above-->
}}
'''Amalie Emmy Noether''', {{IPA-de|ˈnøːtɐ}}, (23 March 1882 – 14 April 1935) was a German [[mathematician]] known for her groundbreaking contributions to [[abstract algebra]] and [[theoretical physics]]. Described by [[Albert Einstein]] and others as the most important woman in the history of mathematics,<ref name="einstein"/><ref name="greatest"/> she revolutionized the theories of [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[algebra over a field|algebras]]. In physics, [[Noether's theorem]] explains the fundamental connection between [[symmetry in physics|symmetry]] and [[conservation law]]s.<ref name="neeman_1999" />

She was born to a Jewish family in the [[Bavaria]]n town of [[Erlangen]]; her father was the mathematician [[Max Noether]]. Emmy 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 dissertation in 1907 under the supervision of [[Paul Albert Gordan|Paul Gordan]] she worked at the Mathematical Institute of Erlangen without pay for seven years. 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]]''.

Noether remained a leading member of the [[Göttingen]] mathematics department until 1933; her students were sometimes called the "Noether boys". In 1924 Dutch mathematician [[Bartel Leendert van der Waerden|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 session|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]]. In 1935 she underwent surgery for an [[ovarian cyst]] and, despite signs of a recovery, died four days later at the age of 53.

Noether's mathematical work has been divided into three "epochs".<ref name=Weyl>{{Harvnb|Weyl|1935}}</ref> In the first (1908–1919), she made significant contributions to the theories of [[algebraic invariant]]s and [[field (mathematics)|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".<ref name="hillled73"/> In the second epoch, (1920–1926), she began work that "changed the face of [abstract] algebra".<ref name="weyl_128" /> In her classic paper ''Idealtheorie in Ringbereichen'' (''Theory of Ideals in Ring Domains'', 1921) Noether developed the theory of [[ideal (ring theory)|ideals]] in [[commutative ring]]s into a powerful tool with wide-ranging applications. She made elegant use of the [[ascending chain condition]], and objects satisfying it are named ''[[Noetherian (disambiguation)|Noetherian]]'' in her honor. In the third epoch, (1927–1935), she published major works on [[noncommutative algebra]]s and [[hypercomplex number]]s and united the [[representation theory]] of [[group (mathematics)|groups]] with the theory of [[module (mathematics)|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]].

==Biography==
[[Image:Erlangen 1916.jpg|thumb|right|Noether grew up in the Bavarian city of [[Erlangen]], depicted here in a 1916 postcard]]
Emmy's father, [[Max Noether]], was descended from a family of wholesale traders in Germany. He had been paralyzed by [[poliomyelitis]] at the age of fourteen. He regained mobility, but one leg remained affected. Largely self taught, he was awarded a [[doctorate]] from the [[University of Heidelberg]] in 1868. After teaching there for seven years, he took a position in the Bavarian city of [[Erlangen]], where he met and married [[Ida Amalia Kaufmann]], the daughter of a prosperous merchant.<ref>{{Harvnb|Kimberling|1981|pp=3–5}}; {{Harvnb|Osen|1974|p=142}}; {{Harvnb|Lederman|Hill|2004|pp=70–71}}; {{Harvnb|Dick|1981|pp=7–9}}.</ref>  Max Noether's mathematical contributions were to [[algebraic geometry]] mainly, following in the footsteps of [[Alfred Clebsch]].  His best known results are the ''[[Brill–Noether theorem]]'' and the residue, or ''[[AF+BG theorem]]'' theorem; several other theorems are associated with him, including ''[[Max Noether's theorem]]''.

Emmy Noether was born on 23 March 1882, the first of four children. Her first name was Amalie, after her mother and paternal grandmother, but she began using her middle name at a young age. As a girl, she was well-liked. She did not stand out academically although she was known for being clever and friendly. Emmy was [[near-sighted]] and talked with a minor [[lisp]] during childhood. A family friend recounted a story years later about young Emmy quickly solving a brain teaser at a children's party, showing logical acumen at that early age.<ref>{{Harvnb|Dick|1981|pp=9–10}}.</ref> Emmy was taught to cook and clean—as were most girls of the time—and she took piano lessons. She pursued none of these activities with passion, although she loved to dance.<ref>{{Harvnb|Dick|1981|pp=10–11}}; {{Harvnb|Osen|1974|p=142}}.</ref>

Of her three brothers, only [[Fritz Noether]], born in 1884, is remembered for his academic accomplishments. After studying in [[Munich]] he made a reputation for himself in [[applied mathematics]]. Her eldest brother, Alfred, was born in 1883, was awarded a doctorate in [[chemistry]] from Erlangen in 1909, but died nine years later. The youngest, Gustav Robert, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928.<ref>{{Harvnb|Dick|1981|pp=25, 45}}; {{Harvnb|Kimberling|p=5}}.</ref>

===University of Erlangen===

[[Image:Paul Albert Gordan 2.jpg|thumb|[[Paul Albert Gordan|Paul Gordan]] supervised Noether's doctoral dissertation on [[invariant (mathematics)|invariants]] of biquadratic forms]]

Emmy 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]]. 

This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing [[coeducation]] would "overthrow all academic order".<ref>Quoted in {{Harvnb|Kimberling|1981|p=10}}.</ref> One of only two women students in a university of 986, Noether was forced to [[academic audit|audit]] classes and required the permission of individual professors whose lectures she wished to attend. Despite the obstacles, on 14 July 1903 she passed the graduation exam at a ''[[Realgymnasium]]'' in [[Nuremberg]].<ref>{{Harvnb|Dick|1981|pp=11–12}}; {{Harvnb|Kimberling|1981|pp=8–10}}; {{Harvnb|Lederman|Hill|2004|p=71}}.</ref>

During the 1903–04 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]]. Soon thereafter, restrictions on women's rights in that university were rescinded. 

Noether returned to Erlangen. She officially reentered the university on 24 October 1904, and declared her intention to focus solely on mathematics. Under the supervision of [[Paul Albert Gordan|Paul Gordan]] she wrote her dissertation, ''Über die Bildung des Formensystems der ternären biquadratischen Form'' (''On Complete Systems of Invariants for Ternary Biquadratic Forms'', 1907). Although it had been well received, Noether later described her thesis as "crap".<ref>{{Harvnb|Kimberling|1981|pp=10–11}}; {{Harvnb|Dick|1981|pp=13–17}}. {{Harvnb|Lederman|Hill|2004|p=71}} write that she completed her doctorate at Göttingen, but this appears to be an error.</ref>  

For the next seven years (1908–1915) she taught at the University of Erlangen's Mathematical Institute without pay, occasionally substituting for her father when he was too ill to lecture.  In 1910 and 1911 she published an extension of her thesis work from three variables to ''n'' variables.

[[Image:Emmy noether postcard 1915.jpg|thumb|250px|right|upright|Noether sometimes used postcards to discuss abstract algebra with her colleague, Ernst Fischer; this card is postmarked 10 April 1915]]

Gordan retired in the spring of 1910, but continued to teach occasionally with his successor, [[Erhard Schmidt]], who left shortly afterward for a position in [[Breslau]].  Gordan retired from teaching altogether in 1911 with the arrival of his second successor, [[Ernst Sigismund Fischer|Ernst Fischer]]. Gordan died in December 1912.  

According to [[Hermann Weyl]], Fischer was an important influence on Noether, in particular by introducing her to the work of [[David Hilbert]]. 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.  This phase marks the beginning of her engagement with [[abstract algebra]], the field of mathematics to which she would make groundbreaking contributions. 

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.<ref>{{Harvnb|Kimberling|1981|pp=11–12}}; {{Harvnb|Dick|1981|pp=18–24}}; {{Harvnb|Osen|1974|p=143}}.</ref>

===University of Göttingen===
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, however, was blocked by the [[Philology|philologists]] and [[historian]]s among the philosophical faculty: women, they insisted, should not become ''[[privatdozent]]''. 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?"<ref name="bath"/> Hilbert responded with indignation, stating, "I do not see that the sex of the candidate is an argument against her admission as ''privatdozent''. After all, we are a university, not a bath house."<ref name="bath">{{Harvnb|Kimberling|1981|p=14}}; {{Harvnb|Dick|1981|p=32}}; {{Harvnb|Osen|1974|pp=144–145}}; {{Harvnb|Lederman|Hill|2004|p=72}}.</ref>

[[Image:Hilbert1912.jpg|thumb|left|upright|In 1915 [[David Hilbert]] invited Emmy Noether to join the mathematics department at the [[University of Göttingen]], challenging the views of some of his colleagues that a woman should not be allowed to teach at a university]]

Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death is unknown. At about the same time Noether's father retired and her brother joined the [[German Army]] to serve in [[World War I]]. She returned to Erlangen for several weeks, mostly to care for her aging father.<ref>{{Harvnb|Dick|1981|pp=24–26}}.</ref>

During her first years teaching at Göttingen she did not have an official position and was not paid; her family paid for her room and board and supported her academic work. Her lectures often were advertised under Hilbert's name, and Noether would provide "assistance".  

Soon after arriving at Gottingen, however, she demonstrated her capabilities by proving the [[theorem]] now known as '''[[Noether's theorem]]''', which shows that a [[conservation law]] is associated with any [[derivative|differentiable]] [[symmetry in physics|symmetry of a physical system]].<ref>{{Harvnb|Osen|1974|pp=144–145}}; {{Harvnb|Lederman|Hill|2004|p=72}}.</ref> 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]]".<ref name="hillled73">{{Harvnb|Lederman|Hill|2004|p=73}}.</ref>

[[Image:Mathematik Göttingen.jpg|thumb|210px|The mathematics department at 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 1918–19]] 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. 

Three years later she received a letter from 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<ref name=Dick188>{{Harvnb|Dick|1981|p=188.}}</ref>). 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 ''Lehrauftrag für Algebra'' a year later.<ref>{{Harvnb|Kimberling|1981|pp=14–18}}; {{Harvnb|Osen|1974|p=145}}; {{Harvnb|Dick|1981|pp=33–34}}.</ref>

===Seminal work in abstract algebra===

Although Noether's theorem had a profound effect upon physics, among mathematicians she is best remembered for her seminal contributions to [[abstract algebra]].  As [[Nathan Jacobson]] says in his Introduction to Noether's ''Collected  Papers'',

<blockquote>
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.
</blockquote>

Noether's groundbreaking work in algebra began in 1920. In collaboration with W. Schmeidler, she then 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]]. The following year she published a landmark paper called, ''Idealtheorie in Ringbereichen'', analyzing [[ascending chain condition]]s with regard to ideals. A noted algebraist, [[Irving Kaplansky]], has called this work "revolutionary",<ref>{{Harvnb|Kimberling|1981|p=18}}</ref> and the publication gave rise to the term "[[Noetherian ring]]" and several other mathematical objects being dubbed, ''[[Noetherian (disambiguation)|Noetherian]]''.<ref>{{Harvnb|Kimberling|1981|p=18}}; {{Harvnb|Dick|1981|pp= 44–45}}; {{Harvnb|Osen|1974|pp=145–146}}</ref>

In 1924, a young Dutch mathematician, [[Bartel Leendert van der Waerden|B. L. 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".<ref>{{Harvnb|van der Waerden|1935|p=100}}.</ref> In 1931 he published ''Moderne Algebra'', a central text in the field; its second volume borrowed heavily from Noether's work. Although Emmy 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".<ref>{{Harvnb|Dick|1981|pp=57–58}}; {{harvnb|Kimberling|1981|p=19}}; {{Harvnb|Lederman|Hill|2004|p=74}}.</ref> She sometimes allowed her colleagues and students to receive credit for her ideas, helping them develop their careers at the expense of her own.<ref>{{Harvnb|Lederman|Hill|2004|p=74}}; {{harvnb|Osen|1974|p=148}}.</ref>

van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which became a major hub of mathematical and physical research. From 1926 to 1930 the Russian [[topology|topologist]], [[Pavel Alexandrov]], lectured at the university, and he and Noether quickly became good friends. 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 only able to help him secure a scholarship from the [[Rockefeller Foundation]].<ref>{{Harvnb|Kimberling|1981|pp=24–25}}; {{Harvnb|Dick|1981|pp=61–63}}.</ref> They met regularly and enjoyed discussions about the intersections of algebra and topology. In his 1935 memorial address, Alexandrov named Emmy Noether "the greatest woman mathematician of all time".<ref>{{Harvnb|Alexandrov|1981|pp=100, 107}}.</ref>

===Lecturing and students===

In Göttingen, Noether supervised more than a dozen doctoral students; her first was [[Grete Hermann]], who defended her dissertation in February 1925. She later spoke reverently of her "dissertation-mother".<ref>{{Harvnb|Dick|1981|p=51}}.</ref> Noether also supervised [[Max Deuring]], who distinguished himself as an undergraduate and went on to contribute significantly to the field of [[arithmetic geometry]]; [[Hans Fitting]], remembered for [[Fitting's theorem]] and the [[Fitting lemma]]; and [[Zeng Jiongzhi]], who proved [[Tsen's theorem]]. She also worked closely 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.<ref>{{Harvnb|Dick|1981|pp=53–57}}.</ref>

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.<ref>{{Harvnb|Dick|1981|pp=37–49}}.</ref> A colleague later described her this way: "Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all."<ref>{{Harvnb|van der Waerden|1935|p=98}}.</ref>

Her frugal lifestyle at first was due to 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]].<ref>{{Harvnb|Dick|1981|pp=46–48}}.</ref> 

Mostly unconcerned about appearance and manners, she focused on her studies to the exclusion of romance and fashion. A distinguished algebraist [[Olga Taussky-Todd]] 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".<ref>{{Harvnb|Taussky|1981|p=80}}</ref> 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 were unable to break through the energetic mathematics discussion she was having with other students.<ref>{{Harvnb|Dick|1981|pp=40–41}}.</ref>

According to van der Waerden's obituary of Emmy Noether, she did not follow a lesson plan for her lectures, which frustrated some students. Instead, she used her lectures as a spontaneous discussion time with her students, to think through and clarify important cutting-edge 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. 

Several of her colleagues attended her lectures, and she allowed some of her ideas, such as the [[crossed product]] (''verschränktes Produkt'' in German) of associative algebras, to be published by others. Noether was recorded as having given at least five semester-long courses at Göttingen:<ref name="scharlau_49">Scharlau, W. "Emmy Noether's Contributions to the Theory of Algebras" in {{Harvnb|Teicher|1999|p=49}}.</ref>

* Winter 1924/25: ''Gruppentheorie und hyperkomplexe Zahlen'' (Group Theory and Hypercomplex Numbers)
* Winter 1927/28: ''Hyperkomplexe Grössen und Darstellungstheorie'' (Hypercomplex Quantities and Representation Theory)
* Summer 1928: ''Nichtkommutative Algebra'' (Noncommutative Algebra)
* Summer 1929: ''Nichtkommutative Arithmetik'' (Noncommutative Arithmetic)
* Winter 1929/30: ''Algebra der hyperkomplexen Grössen'' (Algebra of Hypercomplex Quantities)

These courses often preceded major publications in these areas.

Noether spoke quickly&mdash;reflecting the speed of her thoughts, many said&mdash;and demanded great concentration from her students. Students who disliked her style often felt alienated; one wrote in a notebook with regard to a class that ended at 1:00&nbsp;pm: "It's 12:50, thank God!"<ref>{{Harvnb|Mac Lane|1981|p=77}}; {{Harvnb|Dick|1981|p=37}}.</ref> Some pupils felt that she relied too much on spontaneous discussions. Her most dedicated students, however, relished the enthusiasm with which she approached mathematics, especially since her lectures often built on earlier work they had done together. 

She developed a close circle of colleagues and students who thought along similar lines and tended to exclude those who did not. "Outsiders" who occasionally visited Noether's lectures usually spent only 30&nbsp;minutes 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."<ref>{{Harvnb|Dick|1981|pp=38–41}}.</ref> 

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.<ref>{{Harvnb|Mac Lane|1981|p=71}}</ref> Later, after she had been dismissed by the [[Third Reich]], she invited students into her home to discuss their future plans and mathematical concepts.<ref>{{Harvnb|Dick|1981|p=76}}</ref>

===Moscow===

[[Image:Staruniversity.jpg|thumb|right|Noether taught at the [[Moscow State University]] during the winter of 1928–29]]

In the winter of 1928–29 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]]''.<ref>{{Harvnb|Dick|1981|pp=63–64}}; {{Harvnb|Kimberling|1981|p=26}}; {{Harvnb|Alexandrov|1981|pp=108–110}}.</ref>

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 (1917)|Russian revolution]]. She was especially happy to see [[Soviet]] advancements 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".<ref name="alex106-109">{{Harvnb|Alexandrov|1981|pp=106–109}}.</ref>

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.<ref name="alex106-109"/> Meanwhile her brother, [[Fritz Noether|Fritz]] accepted a position at the Research Institute for Mathematics and Mechanics in [[Tomsk]], in the Siberian Federal District of Russia, after losing his job in Germany.<ref>{{Harvnb|Osen|1974|p=150}}; {{Harvnb|Dick|1981|pp=82–83}}.</ref>

===Recognition===
In 1932 Emmy Noether and [[Emil Artin]] received the [[Ackermann–Teubner Memorial Award]] for their contributions to mathematics.<ref>{{cite web |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Noether_Emmy.html |title=Emmy Amalie Noether |accessdate=2008-09-04  |publisher=dcs.st-and.ac.uk}}</ref> The prize carried a monetary reward of 500&nbsp;[[German Reichsmark|Reichsmarks]] 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''<ref name="1932a">{{Harvnb|Dick|1981|pp=72–73}}; {{Harvnb|Kimberling|1981|pp=26–27}}.</ref> (full professor).<ref name=Dick188/>

[[Image:Zuerich vier Kirchen.jpg|thumb|left|Noether visited [[Zürich]] in 1932 to deliver a plenary address at the [[International Congress of Mathematicians]]]]

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]].<ref>{{Harvnb|Hasse|1933|p=731}}</ref> This pleased her immensely. He also sent her a mathematical riddle, the "mμν-riddle of syllables", which she solved immediately; the riddle has been lost.<ref name="1932a"/>

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 eight hundred people, including Noether's colleagues [[Hermann Weyl]], [[Edmund Landau]], and [[Wolfgang Krull]]. There were four hundred and twenty 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.<ref>{{Harvnb|Kimberling|1981|pp=26–27}}; {{Harvnb|Dick|1981|pp=74–75}}.</ref>

===Expulsion from Göttingen===

When [[Adolf Hitler]] became the [[Chancellor of Germany (German Reich)|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" and was aided by a [[privatdozent]] named Werner Weber, a former student of Emmy Noether. [[Antisemitism|Antisemitic]] attitudes created a climate hostile to Jewish professors.  One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."<ref name="kim29">{{Harvnb|Kimberling|1981|p=29}}.</ref>

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&mdash;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."<ref name="expulsion"/> Several of Noether's colleagues, including [[Max Born]] and [[Richard Courant]], had their positions revoked.<ref name="expulsion">{{Harvnb|Dick|1981|pp=75–76}}; {{Harvnb|Kimberling|1981|pp=28–29}}.</ref> Noether accepted the decision calmly, providing support for others during this difficult time. Hermann Weyl later wrote that "Emmy Noether&mdash;her courage, her frankness, her unconcern about her own fate, her conciliatory spirit&mdash;was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace."<ref name="kim29"/> 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.<ref name="expulsion"/>

===Bryn Mawr===

[[Image:Bryn Mawr Sunset.jpg|thumb|right|[[Bryn Mawr College]] provided a welcoming home for Noether during the last two years of her life]]

As dozens of newly unemployed professors began searching for positions outside of Germany, their colleagues in the United States sought to provide assistance and job opportunities for them. [[Albert Einstein]] and [[Hermann Weyl]] were appointed by the [[Institute for Advanced Study]] in [[Princeton, 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.<ref>{{Harvnb|Dick|1981|pp=78–79}}; {{Harvnb|Kimberling|1981|pp=30–31}}.</ref>

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. Noether in action!"<ref>{{Harvnb|Kimberling|1981|pp=32–33}}; {{Harvnb|Dick|1981|p=80}}.</ref> Noether and a small team of students worked quickly through van der Waerden's 1930 book ''Moderne Algebra I'' and parts of [[Erich Hecke]]'s ''Theorie der algebraischen Zahlen'' (''Theory of algebraic numbers'', 1908).<ref>{{Harvnb|Dick|1981|pp=80–81}}.</ref>

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of [[Abraham Flexner]] and [[Oswald Veblen]]. She also worked with and supervised [[Abraham Adrian Albert|Abraham Albert]] and [[Harry Vandiver]].<ref>{{Harvnb|Dick|1981|pp=81–82}}.</ref> However, she remarked about [[Princeton University]] that she was not welcome at the "men's university, where nothing female is admitted".<ref>{{Harvnb|Dick|1981|p=81}}.</ref> 

Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects.<ref>{{Harvnb|Osen|1974|p=151}}; {{Harvnb|Dick|1981|p=83}}.</ref> In the summer of 1934 she briefly returned to Germany to see Emil Artin and her brother Fritz before he left for Tomsk. Although many of her former colleagues had been forced out of the universities she was able to use the library as a "foreign scholar".<ref>{{Harvnb|Dick|1981|p=82}}; {{Harvnb|Kimberling|1981|p=34}}.</ref>

===Death===

[[Image:Bryn Mawr College Cloisters.JPG|thumb|250px|right|Noether's remains were placed under the walkway surrounding the cloisters of Bryn Mawr's [[Bryn_Mawr_College#M._Carey_Thomas_Library|M. Carey Thomas Library]]]]

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]]".<ref name="tumor"/> 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 recovered quickly from a [[Circulatory system|circulatory]] collapse on the fourth. On 14 April, she fell unconscious, her temperature soared to {{convert|109|°F|°C|sigfig=4}}, 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."<ref name="tumor">{{Harvnb|Kimberling|1981|pp=37–38}}.</ref>

A few days after Noether's death her friends and associates at Bryn Mawr held a small memorial service at President Park's house. Hermann Weyl and [[Richard Brauer]] traveled from Princeton and spoke with Wheeler and Taussky about their departed colleague. In the months which followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, and [[Pavel Alexandrov]] in paying their respects. Her body was cremated and the ashes interred under the walkway around the cloisters of the [[Bryn Mawr College#M. Carey Thomas Library|M. Carey Thomas Library]] at Bryn Mawr.<ref>{{Harvnb|Kimberling|1981|p=39}}.</ref>

==Contributions to mathematics and physics==
First and foremost Noether is remembered as an [[abstract algebra|algebraist]], although her work also had far-ranging consequences for [[theoretical physics]] and [[topology]].  She showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.<ref>{{Harvnb|Osen|1974|pp=148–149}}; {{Harvnb|Kimberling|1981|pp=11–12}}.</ref> Her friend and colleague [[Hermann Weyl]] described her scholarly output in three epochs.<ref>"Emmy Noether’s scientific production
fell into three clearly distinct 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." {{Harv|Weyl|1935}}</ref>
 
In the first epoch (1908–19), Noether dealt primarily with [[invariant theory|differential and algebraic invariants]], beginning with her dissertation under [[Paul Albert 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]].  After moving to Göttingen in 1915, she produced her seminal work for physics, the two [[Noether's theorem]]s.  

In the second epoch (1920–26), Noether devoted herself to developing the theory of [[ring (mathematics)|mathematical rings]].<ref>{{Harvnb|Gilmer|1981|p=131}}.</ref>  

In the third epoch (1927–35), Noether focused on [[noncommutative algebra]], [[linear map|linear transformations]], and commutative number fields.<ref>{{Harvnb|Kimberling|1981|pp=10–23}}.</ref>

===Historical context===

In the century from 1832 to Noether's death in 1935, the field of mathematics&mdash;specifically [[algebra]]&mdash;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 function|cubic]], [[quartic equation|quartic]], and [[quintic equation]]s, as well as on the [[root of unity|related problem]] of constructing [[regular polygon]]s using [[compass and straightedge constructions|compass and straightedge]].  Beginning with [[Carl Friedrich Gauss]]' 1829 proof that [[prime number]]s such as five can be [[integer factorization|factored]] in [[Gaussian integer]]s, [[Évariste Galois]]' introduction of [[group (mathematics)|groups]] in 1832, and [[William Rowan Hamilton]]'s discovery of [[quaternion]]s in 1843, however, 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]].<ref name="algebra_contributions_paramount">G.E. {{Harvnb|Noether|1986|p=168}}.</ref>

====Abstract algebra and ''begriffliche Mathematik'' (conceptual mathematics)====

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 [[Closure_(mathematics)|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 [[associativity|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 adding zero to a number or multiplying it by one), and for every element there must be an [[inverse element]].

A ''ring'' likewise, has a set of elements, but now has ''two'' operations. The first operation must make the set a group, and the second operation is associative and [[distributivity|distributive]] with respect to the first operation.  It may or may not be [[commutativity|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&mdash;the order of the elements does not matter. 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 (mathematics)|field]] is defined as a commutative division ring.

Groups are frequently studied through ''[[group representation]]s''. 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 represents symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. This is a type of symmetry of space, because space itself does not change when it is rotated even though the positions of objects in it do. Noether used these sorts of symmetries in her work on invariants in physics.

A powerful way of studying rings is through their ''[[module (mathematics)|module]]s''. 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. 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 over a field|algebra]]''. (The word algebra means both a subject within mathematics as well as an object studied in the subject of algebra.) 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.

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 example, the elements might be [[word (computing)|computer data words]], where the first combining operation is [[exclusive or]] and the second is [[logical conjunction]]. 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.  As van&nbsp;der&nbsp;Waerden recalled in his obituary of her,<ref>{{Harvnb|Dicke|1981|p=101}}</ref>

<blockquote>
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.
</blockquote>

This is the ''begriffliche Mathematik'' (purely conceptual mathematics) that was characteristic of Noether.  This style of mathematics was adopted by other mathematicians and, after her death, flowered into new forms, such as [[category theory]].

;Integers as an example of a ring

The [[integer]]s form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication.  Any pair of integers can be [[addition|added]] or [[multiplication|multiplied]], always resulting in another integer, and the first operation, addition, is [[commutativity|commutative]], i.e., for any elements a and b in the ring, ''a''&nbsp;+&nbsp;''b'' = ''b''&nbsp;+&nbsp;''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 [[matrix (mathematics)|matrices]] and [[quaternion]]s.   The integers do not form a division ring, because the second operation cannot always be inverted; there is no integer ''a'' such that 3&nbsp;×&nbsp;''a''&nbsp;=&nbsp;1.

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 number]]s. Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem, now called the ''[[Lasker–Noether theorem]]'', for the [[ideal (mathematics)|ideals]] of many rings. 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.

===First epoch (1908–19)===
====Algebraic invariant theory====
[[Image:Emmy Noether - Table of invariants 2.jpg|thumb|250px|right|Table 2 from Noether's dissertation <ref>{{harvnb|Noether|1908}}</ref> on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables ''x'' and ''u''. The horizontal direction of the table lists the invariants with increasing grades in ''x'', while the vertical direction lists them with increasing grades in ''u''.]]

Much of Noether's work in the first epoch of her career was associated with [[invariant theory]], principally [[algebraic invariant theory]]. Invariant theory is concerned with expressions that remain constant (invariant) under a [[group (mathematics)|group]] of transformations. As an everyday example, if a rigid yardstick is rotated, the coordinates (''x'', ''y'', ''z'') of its endpoints change, but its length ''L'' given by the formula {{nowrap|1=''L''<sup>2</sup> = Δ''x''<sup>2</sup> + Δ''y''<sup>2</sup> + Δ''z''<sup>2</sup>}} 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]].
The [[archetype|archetypal]] example of an invariant is the [[discriminant]] ''B''<sup>2</sup> − 4''AC'' of a binary quadratic form ''Ax''<sup>2</sup> + ''Bxy'' + ''Cy''<sup>2</sup>. This is called an invariant because it is unchanged by linear substitutions ''x''→''ax'' + ''by'', ''y''→''cx'' + ''dy'' with determinant ''ad'' − ''bc'' = 1.
These substitutions form the [[special linear group]] ''SL''<sub>2</sub>. (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.) One can ask for all polynomials in ''A'', ''B'', and ''C'' that are unchanged by the action of ''SL''<sub>2</sub>; 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''<sub>0</sub>x<sup>''r''</sup>''y''<sup>0</sup> + ... + ''A''<sub>''r''</sub>x<sup>0</sup>''y''<sup>''r''</sup> of higher degree, which will be certain polynomials in the coefficients
''A''<sub>0</sub>, ... , ''A''<sub>''r''</sub>, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.

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.   Noether's advisor, Paul Albert 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.<ref>M. {{Harvnb|Noether|1914|p=11}}</ref><ref>{{Harvnb|Gordan|1870}}</ref> 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.<ref>{{Harvnb|Weyl|1944|pp=618–621}}</ref><ref>{{Harvnb|Hilbert|1890|p=531}}</ref> Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the [[special orthogonal group]].<ref>{{Harvnb|Hilbert|1890|p=532}}</ref> His first proof caused some controversy because it did not give a method for constructing the  generators, although in later work he made his method constructive.  For her thesis, Noether extended Gordan's computational proof to homogeneous polynomials in three variables. Noether's constructive approach made it possible to study the relationships among the invariants. Later, after she had turned to more abstract methods, Noether called her thesis ''[[:de:Mist|Mist]]'' (crap) and ''Formelngestrüpp'' (a jungle of equations).

====Galois theory====

[[Galois theory]] concerns transformations of [[field (mathematics)|number fields]] that [[permutation|permute]] the roots of an equation. Consider a polynomial equation of a variable ''x'' of [[Degree of a polynomial|degree]] ''n'', in which the coefficients are drawn from some "ground" field, which might be, for example, the field of [[real number]]s, [[rational number]]s, or the [[integer]]s [[modular arithmetic|modulo]]&nbsp;7. There may or may not be choices of ''x'', which make this polynomial evaluate to zero. Such choices, if they exist, are called [[root (mathematics)|root]]s. If the polynomial is ''x''<sup>2</sup>&nbsp;+&nbsp;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. If the field is [[field extension|extended]], however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree. 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, {{nowrap|1=''i''<sup>&nbsp;2</sup> = −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.

The [[Galois group]] of a polynomial is the set of all ways of transforming the splitting field, while preserving the ground field and the roots of the polynomial. (In mathematical jargon, these transformations are called [[automorphism]]s.) The Galois group of {{nowrap|''x''<sup>2</sup> + 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 [[subgroup]]s of the Galois group.

In 1918, Noether published a seminal paper on the [[inverse Galois problem]].<ref>{{harvnb|Noether|1918}}.</ref>  Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "[[Noether's problem]]", which asks whether the fixed field of a subgroup ''G'' of the [[symmetric group|permutation group]] ''S''<sub>''n''</sub> acting on the field {{nowrap|''k''(''x''<sub>1</sub>, ... , ''x''<sub>''n''</sub>)}} always is a pure [[transcendental extension]] of the field ''k''. (She first mentioned this problem in a 1913 paper,<ref>{{harvnb|Noether|1913}}.</ref> where she attributed the problem to her colleague [[Ernst Sigismund Fischer|Fischer]].) She showed this was true for {{nowrap|1=''n'' = 2}}, 3, or 4. In 1969, [[R. G. Swan]] found a counter-example to Noether's problem, with {{nowrap|1=''n'' = 47}} and ''G'' a [[cyclic group]] of order 47<ref>{{harvnb|Swan|1969|p=148}}.</ref> (although this group can be realized as a [[Galois group]] over the rationals in other ways). The inverse Galois problem remains unsolved.<ref>{{Harvnb|Malle|Matzat|1999}}.</ref>

====Physics====
{{main|Noether's theorem|Conservation law|Constant of motion}}

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]], 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, due to the fact that gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern [[theoretical physics]], with her '''first [[Noether's theorem]]''', which she proved in 1915, but did not publish until 1918.<ref>{{harvnb|Noether|1918b}}</ref>  She solved the problem not only for general relativity, but determined the conserved quantities for ''every'' system of physical laws that possesses some continuous symmetry. 

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."<ref>{{Harvnb|Kimberling|1981|p=13}}.</ref>

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.<ref name="ledhill">{{Harvnb|Lederman|Hill|2004|pp=97–116}}.</ref> The physical system itself need not be symmetric; a jagged asteroid tumbling in space [[Angular_momentum#Conservation_of_angular_momentum|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 law]]s of [[momentum|linear momentum]] and [[energy]] within this system, respectively.

Noether's theorem has become a fundamental tool of modern [[theoretical physics]], both because of the insight it gives into conservation laws, and also, as a practical calculation tool.<ref name="neeman_1999">[[Yuval Ne'eman|Ne'eman, Yuval]]. "The Impact of Emmy Noether's Theorems on XX1st Century Physics", {{Harvnb|Teicher|1999|p=83–101}}.</ref> 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.

===Second epoch (1920–26)===

Although the results of Noether's first epoch were impressive and useful, her fame as a mathematician 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.  

In these epochs, she was not merely applying ideas and methods of 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 [[ideal (mathematics)|ideals]] in [[ring (mathematics)|rings]], generalizing  earlier work of [[Richard Dedekind]].  She also is renowned for developing ascending chain conditions, a simple finiteness condition that yielded powerful results in her hands.  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 [[elimination theory]] and the [[algebraic variety|algebraic varieties]] that had been studied by her father.

====Ascending and descending chain conditions====

In this epoch, Noether became famous for her deft use of ascending (''Teilerkettensatz'') or descending (''Vielfachenkettensatz'') chain conditions.  A sequence of [[empty set|non-empty]] [[subset]]s ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, etc. of a [[Set (mathematics)|set]] ''S'' is usually said to be ''strictly ascending'', if each is a subset of the next

:<math>
A_{1} \subset A_{2} \subset A_{3} \subset \cdots
</math>

The [[ascending chain condition]] requires that such sequences break off after a finite number of steps; in other words, all such sequences of subsets must be finite.  Conversely, with ''strictly descending'' sequences of subsets

:<math>
A_{1} \supset A_{2} \supset A_{3} \supset \cdots
</math>

the descending chain condition requires that such sequences break off after a finite number.

Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects&mdash;and, on the surface, they might not seem very powerful.  Noether showed how to exploit such conditions, however, to maximum advantage: for example, how to use them 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.

Many types of objects in [[abstract algebra]] can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called [[Noetherian (disambiguation)|''Noetherian'']] in her honor. 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 (mathematics)|module]] in which every strictly ascending chain of submodules breaks off after a finite number.  A  [[Noetherian space]] is a [[topological space]] in which every strictly increasing chain of open subspaces breaks off after a finite number of terms; this definition is made so that the [[spectrum of a ring|spectrum]] of a Noetherian ring is a Noetherian topological space.

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.

Another application of such chain conditions is in [[Noetherian induction]]&mdash;also known as [[well-founded induction]]&mdash;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 the 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 counterexample, there is a smaller counterexample.

====Commutative rings, ideals, and modules====

Noether's paper, ''Idealtheorie in Ringbereichen'' (''Theory of Ideals in Ring Domains'', 1921),<ref>{{harvnb|Noether|1921}}.</ref> is the foundation of general commutative ring theory, and gives one of the first general definitions of a [[commutative ring]].<ref name="Gil133"/> 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 [[ideal (ring theory)|ideals]], every ideal is finitely generated. In 1943, French mathematician [[Claude Chevalley]] coined the term, ''[[Noetherian ring]]'', to describe this property.<ref name="Gil133">{{Harvnb|Gilmer|1981|p=133}}.</ref>  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 number]]s, and that this decomposition is unique.

Noether's work ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'' (''Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields'', 1927)<ref>{{harvnb|Noether|1927}}.</ref> characterized the rings in which the ideals have unique factorization into prime ideals as the [[Dedekind domain]]s: integral domains that are Noetherian, 0 or 1-[[Krull dimension|dimensional]], and [[Integrality|integrally closed]] in their quotient fields. This paper also contains what now are called the [[isomorphism theorem]]s, which describe some fundamental [[natural isomorphism]]s, and some other basic results on Noetherian and [[Artinian module]]s.

====Elimination theory====

In 1923–24, Noether applied her ideal theory to [[elimination theory]]&mdash;in a formulation that she attributed to her student, Kurt Hentzelt&mdash;showing that fundamental theorems about the  [[polynomial factorization|factorization of polynomials]] could be carried over directly.<ref name="noether_elimination_theory">{{Harvnb|Noether|1923}}, {{Harvnb|Noether|1923b}}, {{Harvnb|Noether|1924}}.</ref>  Traditionally, [[elimination theory]] is concerned with eliminating one or more variables from a system of polynomial equations, usually by the method of [[resultant]]s. For illustration, the system of equations often can be written in the form of a matrix ''M'' (missing the variable ''x'') times a vector ''v'' (having only different powers of ''x'') equaling the zero vector, {{nowrap|1=''M•v'' = ''0''}}. Hence, the [[determinant]] of the matrix ''M'' must be zero, providing a new equation in which the variable ''x'' has been eliminated.

====Invariant theory of finite groups====
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,<ref>{{harvnb|Noether|1915}}</ref> 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 homogenous 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 |''G''|!, the [[factorial]] of the order |''G''| of the group ''G''. The number of generators need not satisfy Noether's bound when the characteristic of the field divides the |''G''|,<ref>{{harvnb|Fleischmann|2000|p=24}}</ref> but Noether was not able to determine whether the bound was correct when the characteristic of the field divides |''G''|! but not |''G''|. For many years, determining the truth or falsity of the bound in this case was an open problem called "Noether's gap". It finally was resolved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.<ref>{{harvnb|Fleischmann|2000|p=25}}, {{harvnb|Fogarty|2001|p=5}}</ref>

In her 1926 paper,<ref>{{harvnb|Noether|1926}}.</ref> 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 [[Haboush's theorem|Mumford conjecture]].<ref>{{Harvnb|Habousch|1975}}.</ref> In this paper Noether also introduced the ''[[Noether normalization lemma]]'', showing that a finitely generated [[integral domain|domain]] ''A'' over a field ''k'' has a set {{nowrap|''x''<sub>1</sub>, ... , ''x''<sub>''n''</sub>}} of [[algebraic independence|algebraically independent]] elements such that ''A'' is [[integrality|integral]] over {{nowrap|''k''[''x''<sub>1</sub>, ... , ''x''<sub>''n''</sub>]}}.

====Contributions to topology====
[[Image:Mug and Torus morph.gif|thumb|right|250px|A continuous deformation ([[homotopy]]) of a coffee cup into a doughnut ([[torus]]) and back]]

As noted by [[Pavel Alexandrov]] and [[Hermann Weyl]] in their obituaries, Noether's contributions to [[topology]] illustrate her  generosity with ideas and how her insights could transform entire fields of mathematics.  In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as their [[connected space|connectedness]].  A common joke is that a topologist can not distinguish her donut from her coffee mug, since they can be smoothly deformed into one another.

Noether is credited with the fundamental ideas that led to the development of [[algebraic topology]] from the earlier [[combinatorial topology]], specifically, the idea of [[Homology_theory#Towards_algebraic_topology|homology group]]s.<ref>{{Harvnb|Hilton|1988|p=284}}</ref>  According to the account of Alexandrov, Noether attended lectures given by [[Heinz Hopf]] and him in the summers of 1926 and 1927, where "she continually made observations, which were often deep and subtle"<ref>{{Harvnb|Dick|1981|p=173}}</ref> and he continues that,

<blockquote>
When ... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly the [[group (mathematics)|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 number]]s, she suggested immediately defining the Betti group as the [[quotient group|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.<ref name="alexandrov_topology_174" >{{Harvnb|Dick|1981|p=174}}</ref>
</blockquote>

Noether's suggestion that topology be studied algebraically, was adopted immediately by Hopf, Alexandrov, and others,<ref name="alexandrov_topology_174" /> and it became a frequent topic of discussion among the mathematicians of Göttingen.<ref name=Hirzebruch>[[Friedrich Hirzebruch|Hirzebruch, Friedrich]]. "Emmy Noether and Topology" in {{Harvnb|Teicher|1999|p=57–61}}.</ref> Noether observed that her idea of a [[Betti group]] makes the [[Euler characteristic|Euler–Poincaré formula]] simple to understand, and Hopf's own work on this subject<ref>{{Harvnb|Hopf|1928}}.</ref> "bears the imprint of these remarks of Emmy Noether".<ref>{{Harvnb|Dick|1981|pp=174–175}}</ref>  Noether mentions her own topology ideas only as an aside in one 1926 publication,<ref>{{Harvnb|Noether|1926b}}</ref> where she cites it as an application of [[group theory]].<ref>Hirzebruch,  Friedrich, "Emmy Noether and Topology" in {{Harvnb|Teicher|1999|p=63}}.</ref>

The algebraic approach to topology was developed independently in [[Austria]]. In a 1926–27 course given in [[Vienna]], [[Leopold Vietoris]] defined a [[homology group]], which was developed by [[Walther Mayer]], into an axiomatic definition in 1928.<ref>Hirzebruch,  Friedrich, "Emmy Noether and Topology" in {{Harvnb|Teicher|1999|p=61–63}}.</ref>

[[Image:Helmut Hasse.jpg|thumb|upright|left|[[Helmut Hasse]] worked with Noether and others to found the theory of [[central simple algebra]]s]]

===Third epoch (1927–35)===
====Hypercomplex numbers and representation theory====

Much work on [[hypercomplex number]]s and [[group representation]]s was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united the results and gave the first general representation theory of groups and algebras.<ref name="noether_1929">{{harvnb|Noether|1929}}.</ref> Briefly, Noether subsumed the structure theory of [[associative algebra]]s and the representation theory of groups into a single arithmetic theory of [[module (mathematics)|modules]] and [[ideal (ring theory)|ideals]] in [[ring (mathematics)|rings]] satisfying [[ascending chain condition]]s. This single work by Noether was of fundamental importance for the development of modern algebra.<ref name="vdWaerden_1985" >{{harvnb|van der Waerden|1985|p=244}}.</ref>

====Noncommutative algebra====

Noether also was responsible for a number of other advancements in the field of algebra. With [[Emil Artin]], [[Richard Brauer]], and [[Helmut Hasse]], she founded the theory of [[central simple algebra]]s.<ref>{{harvnb|Lam|1981|pp=152–153}}.</ref>

A seminal paper by Noether, Helmut Hasse, and [[Richard Brauer]] pertains to [[division algebra]]s,<ref name="hasse_1932" >{{harvnb|Brauer|Hasse|Noether|1932}}.</ref>  which are algebraic systems in which division is possible. They proved two important theorems: a [[Hasse principle|local-global theorem]] stating that if a finite dimensional central division algebra over a [[Algebraic number field|number field]] splits locally everywhere then it splits globally (so is trivial), and from this, deduced their ''Hauptsatz'' ("main theorem"): ''every finite dimensional [[central simple algebra|central]] [[division algebra]] over an [[algebraic number]] [[field (mathematics)|field]] F splits over a [[Abelian extension|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 field]]s.<ref>{{harvnb|Noether|1933}}.</ref> 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]]<ref>{{harvnb|Brauer|Noether|1927}}</ref> gives a characterization of the splitting fields of a central division algebra over a field.

==Assessment, recognition, and memorials==
[[Image:Emmy-noether-campus siegen.jpg|thumb|250px|right|The Emmy Noether Campus at the [[University of Siegen]] is home to its mathematics and physics departments - image by Bob Ionescu]]

Noether's work continues to be relevant for the development of theoretical physics and mathematics and she consistently is ranked as one of the greatest mathematicians of the twentieth century.  In his obituary, fellow algebraist [[Bartel Leendert van der Waerden|B. L. van der Waerden]] says that her mathematical originality was "absolute beyond comparison",<ref>{{Harvnb|Dick|1981|p=100}}</ref> and Hermann Weyl said that Noether "changed the face of [[abstract algebra|algebra]] by her work".<ref name="weyl_128">{{Harvnb|Dick|1981|p=128}}</ref> During her lifetime and even until today, Noether has been characterized as the greatest woman mathematician in recorded history by mathematicians <ref name="greatest">{{Harvnb|Osen|1974|p=152}}; {{Harvnb|Alexandrov|1981|p=100}}.</ref><ref>{{Harvnb|James|2002|p=321}}.</ref> such as [[Pavel Alexandrov]],<ref>{{Harvnb|Dick|1981|p=154}}</ref> [[Hermann Weyl]],<ref>{{Harvnb|Dick|1981|p=152}}</ref> and [[Jean Dieudonné]].<ref name="g_noether_p167">G.E. {{Harvnb|Noether|1987 | p= 167}}.</ref> 

In a letter to ''[[The New York Times]]'', [[Albert Einstein]] wrote:<ref name="einstein">Einstein, Albert. [http://www-history.mcs.st-andrews.ac.uk/Obits2/Noether_Emmy_Einstein.html "Professor Einstein Writes in Appreciation of a Fellow-Mathematician".] 5 May 1935. Online at the [[MacTutor History of Mathematics archive]]. Retrieved on 13 April 2008.</ref>
<blockquote>
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.
</blockquote>

On 2 January 1935, a few months before her death, mathematician [[Norbert Wiener]] wrote that <ref>{{Harvnb|Kimberling|1981|p=35}}</ref>
<blockquote>Miss Noether is ... the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of [[Marie Curie|Madame Curie]].</blockquote>

At an exhibition at the 1964 [[Expo (exhibition)|World's Fair]] devoted to [[Mathematica: A World of Numbers... and Beyond|Modern Mathematicians]], Noether was the only woman represented among the notable mathematicians of the modern world.<ref>Duchin, Moon. [http://www.math.ucdavis.edu/~mduchin/111/readings/genius.pdf "The Sexual Politics of Genius"] (PDF). December 2004. ''University of Chicago''. Retrieved on 13 April 2008.</ref> 

Noether has been honored in several memorials, 
* The [[Association for Women in Mathematics]] holds a [[Noether Lecture]] to honor women in mathematics every year; in its 2005 pamphlet for the event, the Association characterizes Noether as "one of the great mathematicians of her time, someone who worked and struggled for what she loved and believed in. Her life and work remain a tremendous inspiration".<ref>[http://www.awm-math.org/noetherbrochure/Introduction.html "Introduction"]. ''Profiles of Women in Mathematics: The Emmy Noether Lectures''. [[Association for Women in Mathematics]]. 2005. Retrieved on 13 April 2008.</ref> 
* Consistent with her dedication to her students, the [[University of Siegen]] houses its mathematics and physics departments in buildings on ''the Emmy Noether Campus''.<ref>[http://www.uni-siegen.de/uni/campus/wegweiser/emmy.html "Emmy-Noether-Campus"]. ''Universität Siegen''. Retrieved on 13 April 2008.</ref> 
* The German Research Foundation ([[Deutsche Forschungsgemeinschaft]]) operates the ''Emmy Noether Programm'', a scholarship providing funding to promising young post-doctorate scholars in their further research and teaching activities.<ref>[http://www.dfg.de/en/research_funding/promoting_young_researchers/emmy_noether/kompaktdarstellung_emmy_noether.html "Emmy Noether Programme: In Brief"]. ''Research Funding''. [[Deutsche Forschungsgemeinschaft]]. n.d. Retrieved on 5 September 2008.</ref> 
* A street in her hometown, Erlangen, has been named after Emmy Noether and her father, Max Noether. 
* The successor to the secondary school she attended in Erlangen has been renamed as ''the Emmy Noether School''.<ref name="g_noether_p167" /> 

Farther from home, 
* The crater [[Nöther (crater)|Nöther]] on the [[far side of the Moon]] is named after her. 
* The [[7001 Noether]] asteroid also is named for Emmy Noether.<ref>{{Harvnb|Schmadel|2003|p=570}}.</ref><ref>Blue, Jennifer. [http://planetarynames.wr.usgs.gov/ Gazetteer of Planetary Nomenclature]. [[United States Geological Survey|USGS]]. 25 July 2007. Retrieved on 13 April 2008.</ref>

==List of doctoral students==

{| class="wikitable sortable" id="Noether_doctoral_students"
! Date !! Student name !! Dissertation title and English translation !! University !! Publication
|-
| 1911.12.16 || Falckenberg, Hans || Verzweigungen von Lösungen nichtlinearer Differentialgleichungen
{|
| Ramifications of Solutions of Nonlinear Differential Equations<sup>§</sup>
|}
| Erlangen || Leipzig 1912
|-
| 1916.03.04 || Seidelmann, Fritz || Die Gesamtheit der kubischen und biquadratischen Gleichungen mit Affekt bei beliebigem Rationalitätsbereich
{|
| Complete Set of Cubic and Biquadratic Equations with Affect in an Arbitrary Rationality Domain<sup>§</sup>
|}
| Erlangen || Erlangen 1916
|-
| 1925.02.25 || [[Grete Hermann|Hermann, Grete]] || Die Frage der endlich vielen Schritte in der Theorie der Polynomideale unter Benutzung nachgelassener Sätze von Kurt Hentzelt
{|
| The Question of the Finite Number of Steps in the Theory of Ideals of Polynomials using Theorems of the Late Kurt Hentzelt<sup>§</sup>
|}
| Göttingen || Berlin 1926
|-
| 1926.07.14 || Grell, Heinrich || Beziehungen zwischen den Idealen verschiedener Ringe
{|
| Relationships between the Ideals of Various Rings<sup>§</sup>
|}
| Göttingen || Berlin 1927
|-
| 1927 ||Doräte, Wilhelm || Über einem verallgemeinerten Gruppenbegriff
{|
| On a Generalized Conceptions of Groups<sup>§</sup>
|}
| Göttingen || Berlin 1927
|-
| died before defense || Hölzer, Rudolf || Zur Theorie der primären Ringe
{|
| On the Theory of Primary Rings<sup>§</sup>
|}
| Göttingen || Berlin 1927
|-
| 1929.06.12 || Weber, Werner || Idealtheoretische Deutung der Darstellbarkeit beliebiger natürlicher Zahlen durch quadratische Formen
{|
| Ideal-theoretic Interpretation of the Representability of Arbitrary Natural Numbers by Quadratic Forms<sup>§</sup>
|}
| Göttingen || Berlin 1930
|-
| 1929.06.26 || Levitski, Jakob || Über vollständig reduzible Ringe und Unterringe
{|
| On Completely Reducible Rings and Subrings<sup>§</sup>
|}
| Göttingen || Berlin 1931
|-
| 1930.06.18 || [[Max Deuring|Deuring, Max]] || Zur arithmetischen Theorie der algebraischen Funktionen
{|
| On the Arithmetic Theory of Algebraic Functions<sup>§</sup>
|}
| Göttingen || Berlin 1932
|-
| 1931.07.29 || [[Hans Fitting|Fitting, Hans]] || Zur Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nichtkommutativen Gruppen
{|
| On the Theory of Automorphism-Rings of Abelian Groups and Their Analogs in Noncommutative Groups<sup>§</sup>
|}
| Göttingen || Berlin 1933
|-
| 1933.07.27 || [[Ernst Witt|Witt, Ernst]] || Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen
{|
| The Riemann-Roch Theorem and Zeta Function in Hypercomplex Numbers<sup>§</sup>
|}
| Göttingen || Berlin 1934
|-
| 1933.12.06 || [[Zeng Jiongzhi|Tsen, Chiungtze]] || Algebren über Funktionenkörper
{|
| Algebras over Function Fields<sup>§</sup>
|}
| Göttingen || Göttingen 1934
|-
| 1934 || Schilling, Otto || Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper
{|
| On Certain Relationships between the Arithmetic of Hypercomplex Number Systems and Algebraic Number Fields<sup>§</sup>
|}
| Marburg || Braunschweig 1935
|-
| 1935 || Stauffer, Ruth || The construction of a normal basis in a separable extension field
| Bryn Mawr || Baltimore 1936
|-
| 1935 || Vorbeck, Werner || Nichtgaloissche Zerfällungskörper einfacher Systeme
{|
| Non-Galois [[Splitting field|Splitting Fields]] of Simple Systems<sup>§</sup>
|}
| Göttingen || 
|-
| 1936 || Wichmann, Wolfgang || Anwendungen der p-adischen Theorie im Nichtkommutativen Algebren
{|
| Applications of the ''p''-adic Theory in Noncommutative Algebras<sup>§</sup>
|}
| Göttingen || ''Monatshefte für Mathematik und Physik'' (1936) '''44''', 203–224.
|}

==Eponymous mathematical topics==

{{col-begin}}

{{col-3}}
* [[Noetherian (disambiguation)|Noetherian]]
* [[Noetherian group]]
* [[Noetherian ring]]
* [[Noetherian module]]
* [[Noetherian space]]

{{col-3}}

* [[Noetherian induction]]
* [[Noetherian scheme]]
* [[Noether normalization lemma]]
* [[Noether problem]]

{{col-3}}
* [[Noether's theorem]]
* [[Lasker–Noether theorem]]
* [[Skolem–Noether theorem]]
* [[Brauer–Noether theorem]]

{{col-end}}

==Notes==
{{reflist|3}}

==References==
===Selected works by Emmy Noether (in German)===
{{main|List of publications by Emmy Noether}}

*{{citation|last=Noether|first=Emmy|title=Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms)|journal=Journal für die reine und angewandte Mathematik|volume=134|year=1908|pages=23–90 and two tables| url = http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261200}}.
*{{citation|last=Noether|first=Emmy|title=Rationale Funkionenkörper (Rational Function Fields)|journal=J. Ber. D. DMV|volume=22|year=1913|pages=316–319 | url = http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=244058}}.
*{{citation|last=Noether|first=Emmy|year= 1915  |url=http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=461158
|title=Der Endlichkeitssatz der Invarianten endlicher Gruppen (The Finiteness Theorem for Invariants of Finite Groups)
| journal=Mathematische Annalen|volume=77|pages= 89–92|doi=10.1007/BF01456821 }}
*{{citation|last=Noether|first=Emmy|title=Gleichungen mit vorgeschriebener Gruppe (Equations with Prescribed Group)|journal=[[Mathematische Annalen]] |volume=78|year=1918|pages=221–229|doi=10.1007/BF01457099}}.
*{{citation|last = Noether|first= Emmy | year = 1918b | title = Invariante Variationsprobleme (Invariant Variation Problems)| journal = Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse | volume = 1918 | pages = 235–257}}. English translation by M. A. Tavel (1918), {{arxiv|physics|0503066}}.
*{{citation | last=Noether | first=Emmy | title=Idealtheorie in Ringbereichen (The Theory of Ideals in Ring Domains)|format=PDF| url=https://commerce.metapress.com/content/m3457w8h62475473/resource-secured/?target=fulltext.pdf&sid=3y4z44usd0c312452hqwpxm3&sh=www.springerlink.com | year=1921 | journal=Mathematische Annalen | issn=0025-5831 | volume=83 | issue=1}}.
*{{citation | last = Noether | first = Emmy | year = 1923|title = Zur Theorie der Polynomideale und Resultanten | url = http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=362882 | journal = Mathematische Annalen | volume = 88 | pages = 53–79 | doi = 10.1007/BF01448441}}.
*{{citation | last = Noether | first = Emmy | year = 1923b | title = Eliminationstheorie und allgemeine Idealtheorie | url = http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=362964 | journal = Mathematische Annalen | volume = 90 | pages = 229–261 | doi = 10.1007/BF01455443}}.
*{{citation | last = Noether | first = Emmy | year = 1924 | title = Eliminationstheorie und Idealtheorie | url = http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248880 | journal = Jahresbericht der Deutschen Mathematiker-Vereinigung | volume = 33 | pages = 116–120}}.
*{{citation|last=Noether|first=Emmy|title=Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik ''p'' (Proof of the Finiteness of the Invariants of Finite Linear Groups of Characteristic ''p'')|journal=Nachr. Ges. Wiss. Göttingen|pages=28–35|year=1926| url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=63971}}.
*{{citation|last=Noether|first=Emmy|title=Ableitung der Elementarteilertheorie aus der Gruppentheorie (Derivation of the Theory of Elementary Divisor from Group Theory) |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung|volume=34 (Abt. 2)|year=1926b|pages=104 | url = http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248861}}.
*{{citation | last=Noether | first=Emmy |  title=Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number Fields)| url=https://commerce.metapress.com/content/v3t6331n8w244275/resource-secured/?target=fulltext.pdf&sid=zt5psvmwxpvxpbqtyy3riv45&sh=www.springerlink.com | year=1927|format=PDF | journal=Mathematische Annalen | volume=96 | issue=1 | pages=26–61 | doi=10.1007/BF01209152}}.
*{{citation|last=Brauer|first=Richard|last2=Noether|first2=Emmy|author1-link=Richard Brauer |title=Über minimale Zerfällungskörper irreduzibler Darstellungen (On the Minimum Splitting Fields of Irreducible Representations)|journal=Sitz. Ber. D. Preuss. Akad. D. Wiss. |year=1927|pages=221–228}}.
*{{citation |last= Noether|first= Emmy | year = 1929 | title = Hyperkomplexe Grössen und Darstellungstheorie (Hypercomplex Quantities and the Theory of Representations)| journal = Mathematische Annalen | volume = 30 | pages = 641–692|doi=10.1007/BF01187794}}.
*{{citation | last=Brauer|first=Richard|first2=Helmut |last2=Hasse|first3=Emmy |last3=Noether|author2-link=  Helmut Hasse| year = 1932 | title = Beweis eines Hauptsatzes in der Theorie der Algebren (Proof of a Main Theorem in the Theory of Algebras)| journal = Journal für Math. | volume = 167 | pages = 399–404 | url= http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=260847}}.
*{{citation| last=  Noether |first=Emmy | year = 1933 | title = Nichtkommutative Algebren (Noncommutative Algebras)| journal = Mathematische Zeitschrift | volume = 37 | pages = 514–541|doi=10.1007/BF01474591}}.
*{{citation|id={{MR|0703862}}|last=Noether|first= Emmy|title=Gesammelte Abhandlungen (Collected papers)|editor-first=Nathan|editor-last= Jacobson|publisher= Springer-Verlag|place= Berlin-New York|year= 1983|pages= viii, 777 | isbn= 3-540-11504-8 }}.

===Additional sources===
* {{citation|authorlink=Pavel Alexandrov|last=Alexandrov|first=Pavel S.|chapter=In Memory of Emmy Noether|title=Emmy Noether: A Tribute to Her Life and Work|editor=James W. Brewer and Martha K. Smith|place=New York|publisher= Marcel Dekker, Inc.|year=1981|isbn=0-8247-1550-0|pages=99–111}}.
* {{citation|last=Blue|first= Meredith|url=http://mcc1.mccfl.edu/fl_maa/proceedings/2001/blue.pdf|format=PDF|title=Galois Theory and Noether's Problem|place=Thirty-Fourth Annual Meeting|publisher= Florida Section of The Mathematical Association of America|year= 2001}}.
*[[Nina Byers]] (1998) "[http://arxiv.org/abs/physics/9807044 E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws.]" in ''Proceedings of a Symposium on the Heritage of Emmy Noether'', held on 2-4 December, 1996, at the [[Bar-Ilan University]], Israel.
*{{citation|last=Byers|first=Nina|authorlink=Nina Byers|chapter=Emmy Noether|title=Out of the Shadows: Contributions of 20th Century Women to Physics|editor=Nina Byers and Gary Williams|place=Cambridge|publisher=Cambridge University Press|year=2006|isbn=0-5218-2197-5}}.
* {{citation|last=Dick|first= Auguste|title=Emmy Noether: 1882–1935|place= Boston|publisher= Birkhäuser|year= 1981| isbn =3-7643-3019-8}}. Trans. H. I. Blocher.
*{{Citation | last1=Fleischmann | first1=Peter | title=The Noether bound in invariant theory of finite groups | accessdate=2008-06-16 | doi=10.1006/aima.2000.1952    | id={{MathSciNet | id = 1800251}} | year=2000 | journal=Advances in Mathematics | volume=156 | issue=1 | pages=23–32}}
*{{Citation | last1=Fogarty | first1=John | title=On Noether's bound for polynomial invariants of a finite group | url=http://www.ams.org/era/2001-07-02/S1079-6762-01-00088-9/ | accessdate=2008-06-16 | id={{MathSciNet | id = 1826990}} | year=2001 | journal=Electronic Research Announcements of the American Mathematical Society | volume=7 | pages=5–7 | doi=10.1090/S1079-6762-01-00088-9}}
* {{citation|last=Gilmer|first=Robert|chapter=Commutative Ring Theory|title=Emmy Noether: A Tribute to Her Life and Work| editor=James W. Brewer and Martha K. Smith|pages= 131–143|place=New York|publisher= Marcel Dekker, Inc.|year=1981|isbn=0-8247-1550-0}}.
* {{Citation | last1=Gordan | first1=Paul | title=Die simultanen Systeme binärer Formen | language=German | doi=10.1007/BF01444021 | year=1870 | journal=[[Mathematische Annalen]] | volume=2 | issue=2 | pages=227–280}}.
* {{citation|authorlink=William Haboush|first=W. J. |last=Haboush|title=Reductive groups are geometrically reductive|journal=Ann. Of Math. |volume=102|year=1975|pages=67–83|url =http://links.jstor.org/sici?sici=0003-486X%28197507%292%3A102%3A1%3C67%3ARGAGR%3E2.0.CO%3B2-1|doi=10.2307/1970974}}.
* {{citation|last=Hasse|first=Helmut|authorlink=Helmut Hasse|title=Die Struktur der R.&nbsp;Brauerschen Algebrenklassengruppe über einem algebraischen Zahlkörper|language=German|doi=10.1007/BF01448916|journal=Mathematische Annalen|volume=107|year=1933|pages=731–760}}. (German)
* {{citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Ueber die Theorie der algebraischen Formen | language=German | month = December | year=1890 | journal=[[Mathematische Annalen]] | volume=36 | issue=4 | pages=473–534 | doi = 10.1007/BF01208503}}.
*{{citation|last=Hilton|first=Peter|year=1988|title=A Brief, Subjective History of Homology and Homotopy Theory in This Century |journal = Mathematics Magazine|volume=60|issue=5|pages=282–291| url = http://www.jstor.org/stable/2689545?origin=JSTOR-pdf}}
*{{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | url=http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=465901 | title=Eine Verallgemeinerung der Euler-Poincaréschen Formel | language=German | year=1928 | journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse | volume=2 | pages=127–136}}.
*{{citation | authorlink = Ioan James|last=James|first= Ioan | year = 2002 | title = Remarkable Mathematicians from Euler to von Neumann | publisher = Cambridge University Press | location = Cambridge | isbn = 0-521-81777-3 }}.
* {{citation|last=Kimberling|first=Clark|chapter=Emmy Noether and Her Influence|pages=3–61|title=Emmy Noether: A Tribute to Her Life and Work| editor=James W. Brewer and Martha K. Smith|place=New York|publisher= Marcel Dekker, Inc.|year=1981|isbn=0-8247-1550-0}}.
* {{citation|last=Lam|first=Tsit Yuen|chapter=Representation Theory|pages=145–156|title=Emmy Noether: A Tribute to Her Life and Work| editor=James W. Brewer and Martha K. Smith|place=New York|publisher= Marcel Dekker, Inc.|year=1981|isbn=0-8247-1550-0}}.
* {{citation|author1-link=Leon M. Lederman|last1=Lederman|first1= Leon M.|author2-link=Christopher T. Hill|first2=Christopher T.|last2=Hill|title=Symmetry and the Beautiful Universe|place= Amherst|publisher=Prometheus Books|year= 2004|isbn=1-59102-242-8}}.
* {{citation|last=Mac Lane|first=Saunders|authorlink=Saunders Mac Lane|chapter=Mathematics at the University of Göttingen 1831–1933|title=Emmy Noether: A Tribute to Her Life and Work| editor=James W. Brewer and Martha K. Smith|pages= 65–78|place=New York|publisher= Marcel Dekker, Inc.|year=1981|isbn=0-8247-1550-0}}.
* {{Citation | last1=Malle | first1=Gunter | last2=Matzat | first2=Bernd Heinrich | title=Inverse Galois theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-62890-3 | id={{MathSciNet | id = 1711577}} | year=1999}}.
* {{citation | authorlink = Gottfried E. Noether|last=Noether|first= Gottfried E.| year = 1987 | title = Women of Mathematics | editor = Grinstein, L.S. and Campbell, P.J. | publisher = Greenwood press | location = New York | isbn = 0-313-24849-4 }}.
* {{citation|last=Noether|first=Max|authorlink=Max Noether|title=Paul Gordan|journal=Mathematische Annalen|volume=75|issue=1|year= 1914|pages=1–41|doi=10.1007/BF01564521}}.
* {{citation|last=Osen|first=Lynn M.|chapter=Emmy (Amalie) Noether|title=Women in Mathematics|publisher=MIT Press|year=1974|isbn=0-262-15014-X|pages=141–152}}.
* {{citation|last=Schmadel|first= Lutz D.|authorlink=Lutz D. Schmadel|title=Dictionary of Minor Planet Names|edition= 5th revised and enlarged|place= Berlin|publisher= Springer-Verlag|year= 2003|isbn=3-540-00238-3}}.
*{{citation|first=Richard G. |last=Swan| title=Invariant rational functions and a problem of Steenrod|journal=Inventiones Mathematicae |volume=7|year=1969|pages=148–158|doi=10.1007/BF01389798}}.
* {{citation|last=Taussky|first=Olga|authorlink=Olga Taussky-Todd|chapter=My Personal Recollections of Emmy Noether|pages=79–92|title=Emmy Noether: A Tribute to Her Life and Work| editor=James W. Brewer and Martha K. Smith|place=New York|publisher= Marcel Dekker, Inc.|year=1981|isbn=0-8247-1550-0}}.
*{{citation|title=The Heritage of Emmy Noether|first= M. (ed.)|last =Teicher|series=Israel Mathematical Conference Proceedings|publisher= [[Bar-Ilan University]]/[[American Mathematical Society]]/[[Oxford University Press]]|year= 1999|id= {{OCLC|223099225}}|isbn= 978-0198510451}}
* {{citation|authorlink=Bartel Leendert van der Waerden|last=van der Waerden|first= B.L.|title=Nachruf auf Emmy Noether (Obituary of Emmy Noether)|journal=Mathematische Annalen|volume= 111 |year=1935|pages= 469–474|doi=10.1007/BF01472233}}. Reprinted in {{harvnb|Dick|1981}}. (German)
* {{citation |last=van der Waerden|first= B.L.| year = 1985 | title = A History of Algebra: from al-Khwārizmī to Emmy Noether | publisher = Springer-Verlag | location = Berlin | isbn = 0-387-13610-X}}.
* {{citation|first=Hermann|last=Weyl|authorlink=Hermann Weyl|title=Emmy Noether|journal=Scripta Mathematica|volume=3|issue=3|pages=201–220|year=1935}}, reprinted as an appendix to {{Harvtxt|Dick|1981}}.
* {{citation | last1=Weyl | first1=Hermann | title=David Hilbert and his mathematical work | doi=10.1090/S0002-9904-1944-08178-0 | id={{MathSciNet | id = 0011274}} | year=1944 | journal=[[Bulletin of the American Mathematical Society]] | volume=50 | pages=612–654}}.

==External links==
* [http://www.physics.ucla.edu/~cwp/articles/noether.trans/german/emmy235.html "Invariante Variationsprobleme", Nachr. v. d. Ges. d. Wiss. zu Göttingen ] Original paper in German with link to English translation
* "Emmy Noether" in [http://cwp.library.ucla.edu/Phase2/Noether,_Amalie_Emmy@861234567.html CWP at UCLA]
* {{MathGenealogy |id=6967}}
* [http://www.agnesscott.edu/lriddle/women/noether.htm "Emmy Noether", Biographies of Women Mathematicians], [[Agnes Scott College]]
* {{MacTutor Biography|id=Noether_Emmy}}
* [http://www.physikerinnen.de/noetherlebenslauf.html Lebensläufe] {{de icon}} Noether's application for admission to the [[University of Erlangen]] and three [[Curriculum vitae|curricula vitae]], two of which are shown in handwriting, with transcriptions. The first of these is in Emmy Noether's own handwriting.
* [http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=39728 Unpublished] and [http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=261200 published] versions {{de icon}} of Noether's 1908 [[doctoral dissertation]] completed at Erlangen.
* [http://faculty.evansville.edu/ck6/bstud/enmc.html Emmy Noether, Mentors & Colleagues] (photo by Clark Kimberling)
* [http://owpdb.mfo.de/search?term=noether Oberwolfach collection of photos of Noether]
* [http://www.univerlag.uni-goettingen.de/hasse-noether/hasse_noether_web.pdf Correspondence] between Noether and [[Helmut Hasse]], 1925–35

<!-- Metadata: see [[Wikipedia:Persondata]] -->
{{featured article}}

{{Persondata
|NAME              = Noether, Emmy
|ALTERNATIVE NAMES = Noether, Amalie; Emmy Noether
|SHORT DESCRIPTION = Mathematician
|DATE OF BIRTH     = 23 March 1882
|PLACE OF BIRTH    = [[Erlangen]], Germany
|DATE OF DEATH     = 14 April 1935
|PLACE OF DEATH    = [[Bryn Mawr, Pennsylvania]], United States
}}
{{DEFAULTSORT:Noether, Emmy}}
[[Category:1882 births]]
[[Category:1935 deaths]]
[[Category:German mathematicians]]
[[Category:20th-century mathematicians]]
[[Category:Algebraists]]
[[Category:University of Erlangen-Nuremberg alumni]]
[[Category:University of Göttingen faculty]]
[[Category:Bryn Mawr College faculty]]
[[Category:People from the Kingdom of Bavaria]]
[[Category:People from Erlangen]]
[[Category:German Jews]]
[[Category:German-American Jews]]
[[Category:German Jews who emigrated to the United States to escape Nazism]]
[[Category:Women mathematicians]]

[[ar:إيمي نويثر]]
[[bg:Еми Ньотер]]
[[ca:Emmy Noether]]
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[[de:Emmy Noether]]
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[[ko:에미 뇌터]]
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[[he:אמי נתר]]
[[ht:Emmy Noether]]
[[lv:Emmija Nētere]]
[[lt:Emmy Noether]]
[[hu:Emmy Noether]]
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[[ja:エミー・ネーター]]
[[no:Emmy Noether]]
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[[ru:Нётер, Эмми]]
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[[zh:埃米·诺特]]