Noether was born to aJewish family in theFranconian town ofErlangen; her father was the mathematicianMax Noether. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at theUniversity of Erlangen–Nuremberg, where her father lectured. After completing her doctorate in 1907 under the supervision ofPaul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years.[9] At the time, women were largely excluded from academic positions. In 1915, she was invited byDavid Hilbert andFelix Klein to join the mathematics department at theUniversity of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, and she spent four years lecturing under Hilbert's name. Herhabilitation was approved in 1919, allowing her to obtain the rank ofPrivatdozent.[9]
Noether's mathematical work has been divided into three "epochs".[10] In the first (1908–1919), she made contributions to the theories ofalgebraic invariants andnumber fields. Her work on differential invariants in thecalculus of variations,Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics".[11] In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra".[12] In her classic 1921 paperIdealtheorie in Ringbereichen (Theory of Ideals in Ring Domains), Noether developed the theory ofideals incommutative rings into a tool with wide-ranging applications. She made elegant use of theascending chain condition, and objects satisfying it are namedNoetherian in her honor. In the third epoch (1927–1935), she published works onnoncommutative algebras andhypercomplex numbers and united therepresentation theory ofgroups with the theory ofmodules 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 asalgebraic topology.
Noether grew up in the Bavarian city ofErlangen, depicted here in a 1916 postcard.
Amalie Emmy Noether was born on 23 March 1882 inErlangen, Bavaria.[13] She was the first of four children of mathematicianMax Noether and Ida Amalia Kaufmann, both from wealthy Jewish merchant families.[14] Her first name was "Amalie", but she began using her middle name at a young age and invariably continued to do so in her adult life and her publications.[b]
In her youth, Noether did not stand out academically, but was known for being clever and friendly. She wasnear-sighted and talked with a minorlisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at an early age.[15] She was taught to cook and clean, as were most girls of the time, and took piano lessons. She pursued none of these activities with passion, but loved to dance.[16]
Emmy Noether with her brothers Alfred,Fritz, and Robert, before 1918
Noether had three younger brothers. The eldest, Alfred Noether, was born in 1883 and was awarded a doctorate inchemistry from Erlangen in 1909, but died nine years later.[17]Fritz Noether was born in 1884, studied inMunich and made contributions toapplied mathematics.[18] He was likely executed in theSoviet Union in 1941 during theSecond World War.[19] The youngest, Gustav Robert Noether, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928.[20][21]
Noether showed early proficiency in French and English. In early 1900, she took the examination for teachers of these languages and received an overall score ofsehr gut (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at theUniversity of Erlangen–Nuremberg,[22] at which her father was a professor.[23]
This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowingmixed-sex education would "overthrow all academic order".[24] One of just two women in a university of 986 students, Noether was allowed only toaudit classes rather than participate fully, and she required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903, she passed the graduation exam at aRealgymnasium inNuremberg.[22][25][26]
Paul Gordan supervised Noether's doctoral dissertation oninvariants of biquadratic forms.
In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded.[28] Noether returned to Erlangen and officially reentered the university in October 1904, declaring her intention to focus solely on mathematics. She was one of six women in her year (two auditors) and the only woman in her chosen school.[29] Under the supervision ofPaul Gordan, she wrote her dissertation,Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms),[30] in 1907, graduatingsumma cum laude later that year.[31] Gordan was a member of the "computational" school of invariant researchers, and Noether's thesis ended with a list of over 300 explicitly worked-out invariants. This approach to invariants was later superseded by the more abstract and general approach pioneered by Hilbert.[32][33] It had been well received, but Noether later described her thesis and some subsequent similar papers she produced as "crap". All of her later work was in a completely different field.[33][34]
From 1908 to 1915, Noether taught at Erlangen's Mathematical Institute without pay, occasionally substituting for her father,Max Noether, when he was too ill to lecture.[35] She joined theCircolo Matematico di Palermo in 1908 and theDeutsche Mathematiker-Vereinigung in 1909.[36] In 1910 and 1911, she published an extension of her thesis work from three variables ton variables.[37]
Noether sometimes used postcards to discuss abstract algebra with her colleague,Ernst Fischer. This card is postmarked 10 April 1915.
Gordan retired in 1910,[38] and Noether taught under his successors,Erhard Schmidt andErnst Fischer, who took over from the former in 1911.[39] According to her colleagueHermann Weyl and her biographerAuguste Dick, Fischer was an important influence on Noether, in particular by introducing her to the work ofDavid Hilbert.[40][41] 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.[42][43]
From 1913 to 1916, Noether published several papers extending and applying Hilbert's methods to mathematical objects such asfields ofrational functions and theinvariants offinite groups.[44] This phase marked Noether's first exposure toabstract algebra, the field to which she would make groundbreaking contributions.[45]
In Erlangen, Noether advised two doctoral students:[46] Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916.[47][48] Despite Noether's significant role, they were both officially under the supervision of her father. Following the completion of his doctorate, Falckenberg spent time inBraunschweig andKönigsberg before becoming a professor at theUniversity of Giessen[49] while Seidelmann became a professor inMunich.[46]
In early 1915, Noether was invited to return to the University of Göttingen by David Hilbert andFelix Klein. Their effort to recruit her was initially blocked by thephilologists andhistorians among the philosophical faculty, who insisted that women should not becomeprivatdozenten. In a joint department meeting on the matter, one faculty member protested: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?"[50][51] Hilbert, who believed Noether's qualifications were the only important issue and that the sex of the candidate was irrelevant, objected with indignation and scolded those protesting her habilitation. His exact words have not been preserved, but his objection is often said to have included the remark that the university was "not a bathhouse".[40][50][52][53] According toPavel Alexandrov's recollection, faculty members' opposition to Noether was based not just in sexism, but also in their objections to hersocial-democratic political beliefs and Jewish ancestry.[53]
David Hilbert invited Noether to join Göttingen mathematics department in 1915, challenging the views of some of his colleagues that a woman should not teach at a university.
Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death is unknown. At about the same time, Noether's father retired and her brother joined theGerman Army to serve inWorld War I. She returned to Erlangen for several weeks, mostly to care for her aging father.[54]
During her first years teaching at Göttingen, she did not have an official position and was not paid. Her lectures often were advertised under Hilbert's name, and Noether would provide "assistance".[55]
Soon after arriving at Göttingen, she demonstrated her capabilities by proving thetheorem now known asNoether's theorem which shows that aconservation law is associated with any differentiablesymmetry of a physical system.[51][56] The paper,Invariante Variationsprobleme, was presented by a colleague,Felix Klein, on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen.[57][58] Noether presumably did not present it herself because she was not a member of the society.[59] American physicistsLeon M. Lederman andChristopher T. Hill argue in their bookSymmetry and the Beautiful Universe that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development ofmodern physics, possibly on a par with thePythagorean theorem".[11]
The University of Göttingen allowed Noether'shabilitation in 1919, four years after she had begun lecturing at the school.
When World War I ended, theGerman Revolution of 1918–1919 brought a significant change in social attitudes, including more rights for women. In 1919, the University of Göttingen allowed Noether to proceed with herhabilitation (eligibility for tenure). Her oral examination was held in late May, and she successfully delivered herhabilitation lecture in June 1919.[60] Noether became aprivatdozent,[61] and she delivered that fall semester the first lectures listed under her own name.[62] She was still not paid for her work.[55]
Three years later, she received a letter fromOtto Boelitz [de], thePrussian Minister for Science, Art, and Public Education, in which he conferred on her the title ofnicht beamteterausserordentlicher Professor (an untenured professor with limited internal administrative rights and functions).[63] This was an unpaid "extraordinary"professorship, not the higher "ordinary" professorship, which was a civil-service position. It recognized the importance of her work, but still provided no salary. Noether was not paid for her lectures until she was appointed to the special position ofLehrbeauftragte für Algebra (Lecturer for Algebra) a year later.[64][65]
Noether's theorem had a significant effect upon classical and quantum mechanics, but among mathematicians she is best remembered for her contributions toabstract algebra. In his introduction to Noether'sCollected Papers,Nathan Jacobson wrote that
The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her — in published papers, in lectures, and in personal influence on her contemporaries.[66]
Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published a paper about thetheory of ideals in which they definedleft and right ideals in aring.[45]
The following year she published the paperIdealtheorie in Ringbereichen,[67] analyzingascending chain conditions with regards to (mathematical)ideals, in which she proved theLasker–Noether theorem in its full generality. Noted algebraistIrving Kaplansky called this work "revolutionary".[68] The publication gave rise to the termNoetherian for objects which satisfy the ascending chain condition.[68][69]
B. L. van der Waerden (pictured in 1980) was heavily influenced by Noether at Göttingen.
In 1924, a young Dutch mathematician,Bartel Leendert van der Waerden, arrived at the University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. Van der Waerden later said that her originality was "absolute beyond comparison".[70] After returning to Amsterdam, he wroteModerne Algebra, a central two-volume text in the field; its second volume, published in 1931, borrowed heavily from Noether's work.[71] Noether did not seek recognition, but he included as a note in the seventh edition "based in part on lectures byE. Artin and E. Noether".[72][73][74] Beginning in 1927, Noether worked closely withEmil Artin,Richard Brauer, andHelmut Hasse onnoncommutative algebras.[40][71]
Van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which had become a major hub of mathematical and physical research. Russian mathematiciansPavel Alexandrov andPavel Urysohn were the first of several in 1923.[75] Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends.[76] He dubbed herder Noether, usingder as an epithet rather than as the masculine German article.[c][76] She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship toPrinceton University for the 1927–1928 academic year from theRockefeller Foundation.[76][79]
In Göttingen, Noether supervised more than a dozen doctoral students;[46] most were withEdmund Landau and others as she was not allowed to supervise dissertations on her own.[80][81] Her first wasGrete Hermann, who defended her dissertation in February 1925.[82] She is best remembered for her work on the foundations ofquantum mechanics, but her dissertation was considered an important contribution toideal theory.[83][84] Hermann later spoke reverently of her "dissertation-mother".[82]
Around the same time, Heinrich Grell and Rudolf Hölzer wrote their dissertations under Noether. Hölzer died oftuberculosis shortly before his defense.[82][85][86] Grell defended his thesis in 1926 and went on to work at theUniversity of Jena and theUniversity of Halle, before losing his teaching license in 1935 due to accusations of homosexual acts.[46] He was later reinstated and became a professor atHumboldt University in 1948.[46][82]
Noether's other students were Wilhelm Dörnte, who received his doctorate in 1927 with a thesis on groups,[103] Werner Vorbeck, who did so in 1935 with a thesis onsplitting fields,[46] and Wolfgang Wichmann, who did so 1936 with a thesis onp-adic theory.[104] There is no information about the first two, but it is known that Wichmann supported a student initiative that unsuccessfully attempted to revoke Noether's dismissal[105] and died as a soldier on theEastern Front duringWorld War II.[46]
Noether developed a close circle of mathematicians beyond just her doctoral students who shared her approach to abstract algebra and contributed to the field's development,[106] a group often referred to as the Noether school.[107][108] An example of this is her close work withWolfgang Krull, who greatly advancedcommutative algebra with hisHauptidealsatz and hisdimension theory for commutative rings.[109] Another isGottfried Köthe, who contributed to the development of the theory ofhypercomplex quantities using Noether and Krull's methods.[109]
In addition to her mathematical insight, Noether was respected for her consideration of others. She sometimes acted rudely toward those who disagreed with her, but gained a reputation for 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.[110] In Noether's obituary, Van der Waerden described her as
Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all.[70]
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.[111] Later, afterNazi Germany dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts.[112]
Noether's frugal lifestyle was at first due to her being denied pay for her work. 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.[113]
Biographers suggest that she was mostly unconcerned about appearance and manners, focusing on her studies.Olga Taussky-Todd, a distinguished algebraist taught by Noether, described a luncheon during which Noether, wholly engrossed in a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped it off from her dress, completely unperturbed".[114] Appearance-conscious students cringed as she retrieved the handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students once approached her during a break in a two-hour class to express their concern, but they were unable to break through the energetic mathematical discussion she was having with other students.[115]
Noether did not follow a lesson plan for her lectures.[70] She spoke quickly and her lectures were considered difficult to follow by many, includingCarl Ludwig Siegel andPaul Dubreil.[116][117] Students who disliked her style often felt alienated.[118] "Outsiders" who occasionally visited Noether's lectures usually spent only half an hour in the room before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been defeated; he has cleared out."[119]
She used her lectures as a spontaneous discussion time with her students, to think through and clarify important problems in mathematics. Some of her most important results were developed in these lectures, and the lecture notes of her students formed the basis for several important textbooks, such as those of van der Waerden and Deuring.[70] Noether transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her.[120][121]
Several of her colleagues attended her lectures and she sometimes allowed others (including her students) to receive credit for her ideas, resulting in much of her work appearing in papers not under her name.[71][72] Noether was recorded as having given at least five semester-long courses at Göttingen:[122]
Winter 1924–1925:Gruppentheorie und hyperkomplexe Zahlen [Group Theory and Hypercomplex Numbers]
Winter 1927–1928:Hyperkomplexe Grössen und Darstellungstheorie [Hypercomplex Quantities and Representation Theory]
Politics was not central to her life, but Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for theRussian Revolution. She was especially happy to seeSoviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by theBolshevik project. This attitude caused her problems in Germany, culminating in her eviction from apension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess".[126]Hermann Weyl recalled that "During the wild times after theRevolution of 1918," Noether "sided more or less with theSocial Democrats".[40] She was a member of theIndependent Social Democrats from 1919 to 1922, a short-lived splinter party. In the words of logician and historianColin McLarty, "she was not a Bolshevist, but was not afraid to be called one."[127]
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 theSoviet Education Ministry. This proved unsuccessful, but they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union.[126]
In 1932, Emmy Noether andEmil Artin received theAckermann–Teubner Memorial Award for their contributions to mathematics.[71] The prize included a monetary reward of 500 ℛ︁ℳ︁ and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to theGöttingenGesellschaft der Wissenschaften (academy of sciences) and was never promoted to the position ofOrdentlicher Professor[128][129] (full professor).[63]
Noether's colleagues celebrated her fiftieth birthday, in 1932, in typical mathematicians' style.Helmut Hasse dedicated an article to her in theMathematische Annalen, wherein he confirmed her suspicion that some aspects ofnoncommutative algebra are simpler than those ofcommutative algebra, by proving a noncommutativereciprocity law.[130] This pleased her immensely. He also sent her a mathematical riddle, which he called the "mμν-riddle of syllables". She solved it immediately, but the riddle has been lost.[128][129]
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 theInternational Congress of Mathematicians inZürich. The congress was attended by 800 people, including Noether's colleaguesHermann Weyl,Edmund Landau, andWolfgang Krull. There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.[129][131]
WhenAdolf Hitler became theGermanReichskanzler in January 1933,Nazi activity around the country increased dramatically. At the University of Göttingen, the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided byprivatdozent and Noether's former studentWerner Weber.Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students wantAryan mathematics and not Jewish mathematics."[91]
One of the first actions of Hitler's administration was theLaw for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" byserving 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."[132][133] Several of Noether's colleagues, includingMax Born andRichard Courant, also had their positions revoked.[132][133]
Noether accepted the decision calmly, providing support for others during this difficult time.Hermann Weyl later wrote that "Emmy Noether – her courage, her frankness, her unconcern about her own fate, her conciliatory spirit – was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace."[91] Typically, Noether remained focused on mathematics, gathering students in her apartment to discussclass field theory. When one of her students appeared in the uniform of the Naziparamilitary organizationSturmabteilung (SA), she showed no sign of agitation and, reportedly, even laughed about it later.[132][133]
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 andHermann Weyl were appointed by theInstitute for Advanced Study inPrinceton, while others worked to find a sponsor required for legalimmigration. Noether was contacted by representatives of two educational institutions:Bryn Mawr College, in the United States, andSomerville College at theUniversity of Oxford, in England. After a series of negotiations with theRockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.[134][135]
At Bryn Mawr, Noether met and befriendedAnna 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!"[136][137]
During her time at Bryn Mawr, Noether formed a group, sometimes called the Noether girls,[138] of four post-doctoral (Grace Shover Quinn,Marie Johanna Weiss,Olga Taussky-Todd, who all went on to have successful careers in mathematics) and doctoral students (Ruth Stauffer).[139] They enthusiastically worked throughvan der Waerden'sModerne Algebra I and parts ofErich Hecke'sTheorie der algebraischen Zahlen (Theory of algebraic numbers).[140] Stauffer was Noether's only doctoral student in the United States, but Noether died shortly before she graduated.[141] She took her examination withRichard Brauer and received her degree in June 1935,[142] with a thesis concerning separablenormal extensions.[143] After her doctorate, Stauffer worked as a teacher for a short period and as a statistician for over 30 years.[46][142]
Her time in the United States was pleasant, as she was surrounded by supportive colleagues and absorbed in her favorite subjects.[147] In mid-1934, she briefly returned to Germany to see Emil Artin and her brotherFritz.[148] The latter, after having been forced out of his job at theTechnische Hochschule Breslau, had accepted a position at the Research Institute for Mathematics and Mechanics inTomsk, in the Siberian Federal District of Russia.[148]
Many of her former colleagues had been forced out of the universities, but she was able to use the library in Göttingen as a "foreign scholar". Without incident, Noether returned to the United States and her studies at Bryn Mawr.[149][150]
In April 1935, doctors discovered atumor in Noether'spelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered anovarian cyst "the size of a largecantaloupe".[151] Two smaller tumors in heruterus appeared to be benign and were not removed to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from acirculatory collapse on the fourth. On 14 April, Noether fell unconscious, her temperature soared to 109 °F (42.8 °C), and she died. "[I]t is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located." She was 53.[151]
Noether's ashes were placed under the cloistered walkway of Bryn Mawr'sOld Library.
A few days after Noether's death, her friends and associates at Bryn Mawr held a small memorial service at College President Park's house.[152] Hermann Weyl and Richard Brauer both traveled from Princeton and delivered eulogies.[153] In the months that followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, andPavel Alexandrov in paying their respects.[5] Her body was cremated and the ashes interred under the walkway around the cloisters of theOld Library at Bryn Mawr.[154][155]
Noether's work inabstract algebra andtopology was influential in mathematics, whileNoether's theorem has widespread consequences fortheoretical physics anddynamical systems. Noether showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.[42] Her friend and colleagueHermann Weyl described her scholarly output in three epochs:
(1) the period of relative dependence, 1907–1919
(2) the investigations grouped around the general theory of ideals 1920–1926
(3) the study of the non-commutative algebras, their representations by linear transformations, and their application to the study of commutative number fields and their arithmetics
In the first epoch (1907–1919), Noether dealt primarily withdifferential and algebraic invariants, beginning with her dissertation underPaul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work ofDavid Hilbert, through close interactions with a successor to Gordan,Ernst Sigismund Fischer. Shortly after moving to Göttingen in 1915, she proved the twoNoether's theorems, "one of the most important mathematical theorems ever proved in guiding the development of modern physics".[11]
In the second epoch (1920–1926), Noether devoted herself to developing the theory ofmathematical rings.[156] In the third epoch (1927–1935), Noether focused onnoncommutative algebra,linear transformations, and commutative number fields.[157] The results of Noether's first epoch were impressive and useful, but her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B. L. van der Waerden in their obituaries of her.[40][70]
In these epochs, she was not merely applying ideas and methods of the earlier mathematicians; rather, she was crafting new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a completely new theory ofideals inrings, generalizing the earlier work ofRichard Dedekind. She is also renowned for developing ascending chain conditions – a simple finiteness condition that yielded powerful results in her hands.[158] Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as the topics ofalgebraic invariants that had been studied by her father andelimination theory, discussed below.
In the century from 1832 to Noether's death in 1935, the field of mathematics — specificallyalgebra — underwent a profound revolution whose reverberations are still being felt. Mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g.,cubic,quartic, andquintic equations, as well as on therelated problem of constructingregular polygons usingcompass and straightedge. Beginning withCarl Friedrich Gauss's 1832 proof thatprime numbers such as five can befactored inGaussian integers,[159]Évariste Galois's introduction ofpermutation groups in 1832 (because of his death, his papers were published only in 1846, by Liouville),William Rowan Hamilton's description ofquaternions in 1843, andArthur Cayley's more modern definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by ever-more-universal rules. Noether's most important contributions to mathematics were to the development of this new field,abstract algebra.[160]
Background on abstract algebra andbegriffliche Mathematik (conceptual mathematics)
Two of the most basic objects in abstract algebra aregroups andrings:
Agroup consists of a set ofelements 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 beclosed (when applied to any pair of elements of the associated set, the generated element must also be a member of that set), it must beassociative, there must be anidentity element (an element which, when combined with another element using the operation, results in the original element, such as by multiplying a number by one), and for every element there must be aninverse element.[161][162]
Aring likewise, has a set of elements, but now hastwo operations. The first operation must make the set acommutative group, and the second operation isassociative anddistributive with respect to the first operation. It may or may not becommutative; this means that the result of applying the operation to a first and a second element is the same as to the second and first — the order of the elements does not matter.[163] If every non-zero element has amultiplicative inverse (an elementx such thatax =xa = 1), the ring is called adivision ring. Afield is defined as a commutative[f] division ring. For instance, theintegers form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication. Any pair of integers can beadded ormultiplied, always resulting in another integer, and the first operation, addition, iscommutative,i.e., for any elementsa andb in the ring,a +b =b +a. The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning thata combined withb might be different fromb combined witha. Examples of noncommutative rings includematrices andquaternions. The integers do not form a division ring, because the second operation cannot always be inverted; for example, there is no integera such that3a = 1.[164][165]
The integers have additional properties which do not generalize to all commutative rings. An important example is thefundamental theorem of arithmetic, which says that every positive integer can be factored uniquely intoprime numbers.[166] Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem, now called theLasker–Noether theorem, for theideals of many rings.[167] As detailed below, Noether's work included determining what propertiesdo hold for all rings, devising novel analogs of the old integer theorems, and determining the minimal set of assumptions required to yield certain properties of rings.
Groups are frequently studied throughgroup representations.[168] In their most general form, these consist of a choice of group, a set, and anaction 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 avector space, and the group describes thesymmetries of the vector space. For example, there is a group which represents the rigid rotations of space. Rotations are a type of symmetry of space, because the laws of physics themselves do not pick out a preferred direction.[169] Noether used these sorts of symmetries in her work on invariants in physics.[170]
A powerful way of studying rings is through theirmodules. 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.[171]
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 analgebra.[g] 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.[172]
Words such as "element" and "combining operation" are very general, and can be applied to many real-world and abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For instance, the elements might be logical propositions, where the first combining operation isexclusive or and the second islogical conjunction.[173] Theorems of abstract algebra are powerful because they are general; they govern many systems. It might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay Noether's gift to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. In his obituary of Noether, van der Waerden recalled that
The maxim by which Emmy Noether was guided throughout her work might be formulated as follows: "Any relationships between numbers, functions, and operations become transparent, generally applicable, and fully productive only after they have been isolated from their particular objects and been formulated as universally valid concepts."[174]
This is thebegriffliche Mathematik (purely conceptual mathematics) that was characteristic of Noether. This style of mathematics was consequently adopted by other mathematicians, especially in the (then new) field of abstract algebra.[175]
Table 2 from Noether's dissertation[30] on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variablesx andu. The horizontal direction of the table lists the invariants with increasing grades inx, while the vertical direction lists them with increasing grades inu.
Much of Noether's work in the first epoch of her career was associated withinvariant theory, principallyalgebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under agroup of transformations.[176] As an everyday example, if a rigidmetre-stick is rotated, the coordinates of its endpoints change, but its length remains the same. A more sophisticated example of aninvariant is thediscriminantB2 − 4AC of a homogeneous quadratic polynomialAx2 +Bxy +Cy2, wherex andy areindeterminates. The discriminant is called "invariant" because it is not changed by linear substitutionsx →ax +by andy →cx +dy with determinantad −bc = 1. These substitutions form thespecial linear groupSL2.[177]
One can ask for all polynomials inA,B, andC that are unchanged by the action ofSL2; these turn out to be the polynomials in the discriminant.[178] More generally, one can ask for the invariants ofhomogeneous polynomialsA0xry0 + ... +Arx0yr of higher degree, which will be certain polynomials in the coefficientsA0, ...,Ar, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.[179]
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, calledgenerators, and then, adding or multiplying the generators together.[180] For example, the discriminant gives a finite basis (with one element) for the invariants of a quadratic polynomial.[178]
Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables.[181][182] 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.[183][184] Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as thespecial orthogonal group.[185]
Noether followed Gordan's lead, writing her doctoral dissertation and several other publications on invariant theory. She extended Gordan's results and also built upon Hilbert's research. Later, she would disparage this work, finding it of little interest and admitting to forgetting the details of it.[186] Hermann Weyl wrote,
[A] greater contrast is hardly imaginable than between her first paper, the dissertation, and her works of maturity; for the former is an extreme example of formal computations and the latter constitute an extreme and grandiose example of conceptual axiomatic thinking in mathematics.[187]
Galois theory concerns transformations ofnumber fields thatpermute the roots of an equation.[188] Consider a polynomial equation of a variablex ofdegreen, in which the coefficients are drawn from someground field, which might be, for example, the field ofreal numbers,rational numbers, or theintegersmodulo 7. There may or may not be choices ofx, which make this polynomial evaluate to zero. Such choices, if they exist, are calledroots.[189] For example, if the polynomial isx2 + 1 and the field is the real numbers, then the polynomial has no roots, because any choice ofx makes the polynomial greater than or equal to one.[190] If the field isextended then the polynomial may gain roots,[191] and if it is extended enough, then it always has a number of roots equal to its degree.[192]
Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots,+i and−i, wherei is theimaginary unit, that is,i 2 = −1. More generally, the extension field in which a polynomial can be factored into its roots is known as thesplitting field of the polynomial.[193]
TheGalois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial.[194] (These transformations are calledautomorphisms.) The Galois group ofx2 + 1 consists of two elements: The identity transformation, which sends every complex number to itself, andcomplex 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, so transformation determines apermutation of then roots among themselves. The significance of the Galois group derives from thefundamental 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 thesubgroups of the Galois group.[195]
In 1918, Noether published a paper on theinverse Galois problem.[196] 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 subgroupG of thepermutation groupSn acting on the fieldk(x1, ...,xn) always is a puretranscendental extension of the fieldk. (She first mentioned this problem in a 1913 paper,[197] where she attributed the problem to her colleagueFischer.) She showed this was true forn = 2, 3, or 4. In 1969,Richard Swan found a counter-example to Noether's problem, withn = 47 andG acyclic group of order 47[198] (although this group can be realized as aGalois group over the rationals in other ways). The inverse Galois problem remains unsolved.[199]
Noether was brought toGöttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understandinggeneral relativity,[200] a geometrical theory ofgravitation developed mainly byAlbert Einstein. Hilbert had observed that theconservation of energy seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of moderntheoretical physics, in a 1918 paper.[201] This paper presented two theorems, of which the first is known asNoether's theorem.[202] Together, these theorems not only solve the problem for general relativity, but also determine the conserved quantities forevery system of physical laws that possesses some continuous symmetry.[203] 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.[204]
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 theangular momentum of the system must be conserved.[170][205] The physical system itself need not be symmetric; a jagged asteroid tumbling in spaceconserves angular momentum despite its asymmetry. Rather, the symmetry of thephysical laws governing the system is responsible for the conservation law. As another example, if a physical experiment works the same way at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for theconservation laws oflinear momentum andenergy within this system, respectively.[206][207]
At the time, physicists were not familiar withSophus Lie's theory ofcontinuous groups, on which Noether had built. Many physicists first learned of Noether's theorem from an article byEdward Lee Hill that presented only a special case of it. Consequently, the full scope of her result was not immediately appreciated.[208] During the latter half of the 20th century, Noether's theorem became a fundamental tool of moderntheoretical physics, because of the insight it gives into conservation laws, and also as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon: If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.[8]
In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions.[158] A sequence ofnon-emptysubsetsA1,A2,A3, ... of asetS is usually said to beascending if each is a subset of the next:
Conversely, a sequence of subsets ofS is calleddescending if each contains the next subset:
A chainbecomes constant after a finite number of steps if there is ann such that for allm ≥n. A collection of subsets of a given set satisfies theascending chain condition if every ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.[209] Chain conditions can be used to show that every set of sub-objects has a maximal/minimal element, or that a complex object can be generated by a smaller number of elements.[210]
Many types of objects inabstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are calledNoetherian in her honor.[211] By definition, aNoetherian ring satisfies an ascending chain condition on its left and right ideals, whereas aNoetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. ANoetherian module is amodule in which every strictly ascending chain of submodules becomes constant after a finite number of steps.[212][213] ANoetherian space is atopological space whose open subsets satisfy the ascending chain condition;[h] this definition makes thespectrum of a Noetherian ring a Noetherian topological space.[214][215]
The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are Noetherian; and,mutatis mutandis, the same holds for submodules and quotient modules of a Noetherian module.[216] 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 thering of formal power series over a Noetherian ring.[217]
Another application of such chain conditions is inNoetherian induction – also known aswell-founded induction – which is a generalization ofmathematical induction. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose thatS is apartially ordered set. One way of proving a statement about the objects ofS is to assume the existence of acounterexample and deduce a contradiction, thereby proving thecontrapositive of the original statement. The basic premise of Noetherian induction is that every non-empty subset ofS contains a minimal element. In particular, the set of all counterexamples contains a minimal element, theminimal counterexample. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.[218]
Noether's paper,Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921),[67] is the foundation of general commutativering theory, and gives one of the first general definitions of acommutative ring.[i][219] 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 onideals, every ideal is finitely generated. In 1943, French mathematicianClaude Chevalley coined the termNoetherian ring to describe this property.[219] A major result in Noether's 1921 paper is theLasker–Noether theorem, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings.[45][220] The Lasker–Noether theorem can be viewed as a generalization of thefundamental theorem of arithmetic which states that any positive integer can be expressed as a product ofprime numbers, and that this decomposition is unique.[167]
Noether's workAbstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields, 1927)[221] characterized the rings in which the ideals have unique factorization into prime ideals (now calledDedekind domains).[222] Noether showed that these rings were characterized by five conditions: they must satisfy the ascending and descending chain conditions, they must possess a unit element, but nozero divisors, and they must beintegrally closed in their associated field of fractions.[222][223] This paper also contains what now are called theisomorphism theorems,[224] which describe some fundamentalnatural isomorphisms, and some other basic results on Noetherian andArtinian modules.[225]
In 1923–1924, Noether applied her ideal theory toelimination theory in a formulation that she attributed to her student, Kurt Hentzelt. She showed that fundamental theorems about thefactorization of polynomials could be carried over directly.[226][227][228]
Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, often by the method ofresultants.[229]For illustration, a system of equations often can be written in the form
Mv = 0
where a matrix (orlinear transform)M (without the variablex) times a vectorv (that only has non-zero powers ofx) is equal to the zero vector,0. Hence, thedeterminant of the matrixM must be zero, providing a new equation in which the variablex has been eliminated.
Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper,[230] Noether found a solution to the finite basis problem for a finite group of transformationsG acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is calledNoether's bound. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field iscoprime to (thefactorial of the order of the groupG). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number,[231] but Noether was not able to determine whether this bound was correct when the characteristic of the field divides but not. For many years, determining the truth or falsehood of this bound for this particular case was an open problem, called "Noether's gap". It was finally solved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.[232][233]
In her 1926 paper,[234] 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 byWilliam Haboush to all reductive groups by his proof of theMumford conjecture.[235] In this paper Noether also introduced theNoether normalization lemma, showing that a finitely generateddomainA over a fieldk has a set{x1, ...,xn} ofalgebraically independent elements such thatA isintegral overk[x1, ...,xn].
A continuous deformation (homotopy) of a coffee cup into a doughnut (torus) and back
As noted byHermann Weyl in his obituary, Noether's contributions totopology illustrate her generosity with ideas and how her insights could transform entire fields of mathematics.[40] In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as theirconnectedness. An old joke is that "a topologist cannot distinguish a donut from a coffee mug", since they can becontinuously deformed into one another.[236]
Noether is credited with fundamental ideas that led to the development ofalgebraic topology from the earliercombinatorial topology, specifically, the idea ofhomology groups.[237] According to Alexandrov, Noether attended lectures given by him andHeinz Hopf in 1926 and 1927, where "she continually made observations which were often deep and subtle"[238] and he continues that,
When ... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly thegroups of algebraic complexes and cycles of a given polyhedron and thesubgroup of the cycle group consisting of cycles homologous to zero; instead of the usual definition ofBetti numbers, she suggested immediately defining the Betti group as thecomplementary (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.[239]
Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others,[239] and it became a frequent topic of discussion among the mathematicians of Göttingen.[240] Noether observed that her idea of aBetti group makes theEuler–Poincaré formula simpler to understand, and Hopf's own work on this subject[241] "bears the imprint of these remarks of Emmy Noether".[242] Noether mentions her own topology ideas only as an aside in a 1926 publication,[243] where she cites it as an application ofgroup theory.[244]
Much work onhypercomplex numbers andgroup representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these earlier results and gave the first general representation theory of groups and algebras.[246][247] This single work by Noether was said to have ushered in a new period in modern algebra and to have been of fundamental importance for its development.[248]
A paper by Noether, Hasse, and Brauer pertains todivision algebras,[250] which are algebraic systems in which division is possible. They proved two important theorems: alocal-global theorem stating that if a finite-dimensional central division algebra over anumber field splits locally everywhere then it splits globally (so is trivial), and from this, deduced theirHauptsatz ("main theorem"):
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 algebraD aresplitting fields.[251] This paper also contains theSkolem–Noether theorem, which states that any two embeddings of an extension of a fieldk into a finite-dimensional central simple algebra overk are conjugate. TheBrauer–Noether theorem[252] gives a characterization of the splitting fields of a central division algebra over a field.[253]
The Emmy–Noether–Campus at theUniversity of Siegen is home to its mathematics and physics departments.[254]Noether is one of the carved stone busts displayed in Germany'sRuhmeshalle München (or Munich Hall of Fame in English).
Noether's work continues to be relevant for the development of theoretical physics and mathematics, and she is considered one of the most important mathematicians of the twentieth century.[155][255] During her lifetime and even until today, Noether has also been characterized as the greatest woman mathematician in recorded history[6][7][256] by mathematicians such asPavel Alexandrov,[257]Hermann Weyl,[258] andJean Dieudonné.[259]
In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematicalgenius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.
In his obituary, fellow algebraistB. L. van der Waerden says that her mathematical originality was "absolute beyond comparison",[260] and Hermann Weyl said that Noether "changed the face of [abstract] algebra by her work".[12] Mathematician and historianJeremy Gray wrote that any textbook on abstract algebra bears the evidence of Noether's contributions: "Mathematicians simply do ring theory her way."[211] Several thingsnow bear her name, including many mathematical objects,[211] and an asteroid, 7001 Noether.[261]
^abEmmy is theRufname, the second of two official given names, intended for daily use. This can be seen in the résumé submitted by Noether to theUniversity of Erlangen–Nuremberg in 1907.[1][2] SometimesEmmy is mistakenly reported as a short form forAmalie, or misreported asEmily; for example, the latter was used byLee Smolin in a letter forThe Reality Club.[3]
^The nickname was not always used in a well-meaning manner.[77] In Noether's obituary, Hermann stated that
The power of your genius seemed to transcend the bounds of your sex, which is why we in Göttingen, in awed mockery, often spoke of you in the masculine form as "der Noether."[40][78]
^When Noether was forced to leave Germany in 1933, she wished for the university to appoint Deuring as her successor,[94] but he only started teaching there in 1950.[93]
^Accounts of Tsen's date of death vary:Kimberling (1981, p. 41) states that he died "some time in 1939 or 40" andDing, Kang & Tan (1999) state that he died in November 1940, but a local newspaper recorded his date of death as 1 October 1940.[101]
^Or whose closed subsets satisfy the descending chain condition.[214]
^The first definition of an abstract ring was given byAbraham Fraenkel in 1914, but the definition in current use was initially formulated by Masazo Sono in a 1917 paper.[219]
^Dick 1981, pp. 51–53. See p. 51: "... Grete Hermann who took her examinations in February 1925 with E. Noether and E. Landau, See also pp. 52–53: "In 1929 Werner Weber obtained a doctor's degree ... The reviewers were E. Landau and E. Noether." Also on p. 53: "He was followed two weeks later by Jakob Levitzki ... who also was examined by Noether and Landau.
^"十月份甯屬要聞" [Main news of Ningshu in October],新寧遠月刊 Xin Ningyuan Yuekang [New Ningyuan Monthly] (in Chinese), vol. 1, no. 3,Xichang,Xikang, 25 November 1940, p. 51,一日 國立西康技藝專科學校教授曾烱之博士在西康衞生院病逝. [1st: Dr. Chiungtze Tsen, professor at National Xikang Institute of Technology, died from illness in Xikang Health Center.]
^abRowe & Koreuber 2020, pp. 27–30. See p. 27: "In 1921, Noether published her famous paper ... [which] dealt with rings whose ideals satisfy the ascending chain condition". See p. 30: "The role of chain conditions in abstract algebra begins with her now classic paper [1921] and culminates with the seminal study [1927]". See p. 28 on strong initial support for her ideas in the 1920s by Pavel Alexandrov and Helmut Hasse, despite "considerable skepticism" from French mathematicians.
^Gauss, Carl F. (1832), "Theoria residuorum biquadraticorum – Commentatio secunda",Comm. Soc. Reg. Sci. Göttingen (in Latin),7:1–34 Reprinted inWerke [Complete Works of C.F. Gauss], Hildesheim:Georg Olms Verlag, 1973, pp. 93–148
^Baez, John C. (2022), "Getting to the Bottom of Noether's Theorem", in Read, James; Teh, Nicholas J. (eds.),The Philosophy and Physics of Noether's Theorems, Cambridge University Press, pp. 66–99,arXiv:2006.14741,ISBN9781108786812
^Klappenecker, Andreas (Fall 2008),"Noetherian induction"(PDF),CPSC 289 Special Topics on Discrete Structures for Computing (Lecture notes),Texas A&M University,archived(PDF) from the original on 4 July 2024, retrieved14 January 2025
Gilmer, Robert (1981), "Commutative Ring Theory", in Brewer, James W.; Smith, Martha K. (eds.),Emmy Noether: A Tribute to Her Life and Work, New York:Marcel Dekker, pp. 131–143,ISBN978-0-8247-1550-2
Kimberling, Clark (1981), "Emmy Noether and Her Influence", in Brewer, James W.; Smith, Martha K. (eds.),Emmy Noether: A Tribute to Her Life and Work, New York:Marcel Dekker, pp. 3–61,ISBN978-0-8247-1550-2
Lam, Tsit Yuen (1981), "Representation Theory", in Brewer, James W.; Smith, Martha K. (eds.),Emmy Noether: A Tribute to Her Life and Work, New York:Marcel Dekker, pp. 145–156,ISBN978-0-8247-1550-2
Mac Lane, Saunders (1981), "Mathematics at the University of Göttingen 1831–1933", in Brewer, James W.; Smith, Martha K. (eds.),Emmy Noether: A Tribute to Her Life and Work, New York:Marcel Dekker, pp. 65–78,ISBN978-0-8247-1550-2
Merzbach, Uta C. (1983), "Emmy Noether: Historical Contexts", in Srinivasan, Bhama; Sally, Judith D. (eds.),Emmy Noether in Bryn Mawr: Proceedings of a Symposium Sponsored by the Association for Women in Mathematics in Honor of Emmy Noether's 100th Birthday, New York, NY: Springer, pp. 161–171,doi:10.1007/978-1-4612-5547-5_12,ISBN978-1-4612-5547-5
Roquette, Peter (2005),The Brauer–Hasse–Noether theorem in historical perspective, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, vol. 15,CiteSeerX10.1.1.72.4101,MR2222818,Zbl1060.01009
Taussky, Olga (1981), "My Personal Recollections of Emmy Noether", in Brewer, James W.; Smith, Martha K (eds.),Emmy Noether: A Tribute to Her Life and Work, New York:Marcel Dekker, pp. 79–92,ISBN978-0-8247-1550-2
Weber, Werner (December 1930), "Idealtheoretische Deutung der Darstellbarkeit beliebiger natürlicher Zahlen durch quadratische Formen" [Ideal-theoretic Interpretation of the Representability of Arbitrary Natural Numbers by Quadratic Forms],Mathematische Annalen (in German),102 (1):740–767,doi:10.1007/BF01782375
Wichmann, Wolfgang (1936), "Anwendungen der p-adischen Theorie im Nichtkommutativen" [Applications of the p-adic Theory in Noncommutative Algebras],Monatshefte für Mathematik (in German),44 (1):203–224,doi:10.1007/BF01699316
——— (1918c),"Invariante Variationsprobleme" [Invariant Variation Problems],Nachr. D. König. Gesellsch. D. Wiss. (in German),918, Göttingen:235–257, archived fromthe original on 5 July 2008 Original German image with link to Tavel's English translation
Blue, Meredith (2001),Galois Theory and Noether's Problem(PDF), 34th Annual Meeting of the Mathematical Association of America,MAA Florida Section, archived fromthe original(PDF) on 29 May 2008, retrieved9 June 2018
.jpg)
.jpg)
