Emmy Noether

Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician renowned for her significant contributions to abstract algebra. She also established Noether's first and second theorems, which are foundational in mathematical physics. Prominent mathematicians including Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener characterized Noether as the most pivotal female figure in the history of mathematics. As a preeminent mathematician of her era, she formulated theories concerning rings, fields, and algebras. In the realm of physics, Noether's theorem elucidates the intrinsic relationship between symmetry and conservation laws.

Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental in mathematical physics. Noether was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl, and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

Noether was born into a Jewish family in Erlangen, a Franconian town; her father, Max Noether, was also a mathematician. Initially, she intended to pursue a career teaching French and English, having passed the requisite examinations; however, she ultimately chose to study mathematics at the University of Erlangen–Nuremberg, where her father held a lecturing position. Following the completion of her doctorate in 1907, supervised by Paul Gordan, she spent seven years working unpaid at the Mathematical Institute of Erlangen. During this period, women were generally barred from holding academic appointments. In 1915, David Hilbert and Felix Klein extended an invitation for her to join the mathematics department at the University of Göttingen, a globally recognized hub for mathematical research. The philosophical faculty raised objections, leading her to lecture for four years under Hilbert's name. Her habilitation was approved in 1919, which enabled her to achieve the rank of Privatdozent.

Noether maintained a prominent role within the Göttingen mathematics department until 1933; her students were occasionally referred to as the "Noether Boys." In 1924, the Dutch mathematician B. L. van der Waerden became part of her academic group and rapidly emerged as a primary interpreter of Noether's concepts; her research formed the basis for the second volume of his influential 1931 textbook, Moderne Algebra. Her algebraic expertise gained global recognition by the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich. The subsequent year, Germany's Nazi government expelled Jewish academics from university posts, prompting Noether to relocate to the United States for a position at Bryn Mawr College in Pennsylvania. At Bryn Mawr, she instructed graduate and post-doctoral female students, notably Marie Johanna Weiss and Olga Taussky-Todd. Concurrently, she delivered lectures and conducted research at the Institute for Advanced Study in Princeton, New Jersey.

Noether's mathematical contributions are categorized into three distinct "epochs." During the first epoch (1908–1919), she advanced the theories of algebraic invariants and number fields. Her research on differential invariants within the calculus of variations, known as Noether's theorem, has been lauded as "one of the most significant mathematical theorems ever established in directing the evolution of modern physics." In the second epoch (1920–1926), she initiated work that "transformed the landscape of [abstract] algebra." In her seminal 1921 paper, Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains), Noether advanced the theory of ideals in commutative rings, transforming it into a widely applicable tool. She masterfully employed the ascending chain condition, and mathematical objects fulfilling this condition are designated as Noetherian in tribute to her. During the third epoch (1927–1935), she published research on noncommutative algebras and hypercomplex numbers, integrating the representation theory of groups with the theory of modules and ideals. Beyond her personal publications, Noether generously shared her insights and is recognized for inspiring several research directions pursued by other mathematicians, even in areas distant from her primary focus, such as algebraic topology.

Biography

Early Life

Amalie Emmy Noether was born on 23 March 1882, in Erlangen, Bavaria. She was the eldest of four children born to mathematician Max Noether and Ida Amalia Kaufmann, both originating from affluent Jewish merchant families. Although her given first name was "Amalie," she adopted her middle name from an early age and consistently used it throughout her adult life and in her published works.

In her youth, Noether did not achieve academic distinction but was recognized for her intellect and amiable disposition. She experienced myopia and a minor lisp during her childhood. A family acquaintance later recounted an anecdote from Noether's youth, illustrating her early logical acumen through the swift resolution of an intellectual puzzle at a children's gathering. She received instruction in domestic skills, a common practice for girls of her era, and took piano lessons. While she pursued none of these activities with particular fervor, she evinced a strong fondness for dancing.

Noether had three younger brothers. The eldest, Alfred Noether, born in 1883, earned a doctorate in chemistry from Erlangen in 1909 but passed away nine years later. Fritz Noether, born in 1884, studied in Munich and contributed to the field of applied mathematics. He was likely executed in the Soviet Union in 1941 during the Second World War. The youngest, Gustav Robert Noether, born in 1889, suffered from chronic illness and died in 1928; details regarding his life are scarce.

Education

Noether demonstrated early aptitude in both French and English. In early 1900, she sat for the examination for language teachers, achieving an overall evaluation of sehr gut (very good). Although this performance rendered her eligible to instruct languages at girls' schools, she opted instead to pursue further academic endeavors at the University of Erlangen–Nuremberg, where her father held a professorship.

This constituted an unorthodox choice; two years prior, the university's Academic Senate had asserted that coeducational instruction would "overthrow all academic order." As one of only two women among 986 students, Noether was permitted solely to audit courses, precluding full participation, and necessitated obtaining individual consent from professors whose lectures she wished to attend. Notwithstanding these impediments, she successfully passed the graduation examination at a Realgymnasium in Nuremberg on July 14, 1903.

During the winter semester of 1903–1904, she undertook studies at the University of Göttingen, participating in lectures delivered by astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert.

In 1903, limitations on women's full matriculation in Bavarian universities were lifted. Noether returned to Erlangen, formally re-enrolling in the university in October 1904 and articulating her exclusive dedication to mathematics. She was one of six women in her cohort (including two auditors) and the sole woman in her chosen academic department. Under the supervision of Paul Gordan, she completed her doctoral dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms), in 1907, achieving graduation with summa cum laude honors later that year. Gordan, a proponent of the "computational" school of invariant theory, oversaw a thesis that concluded with an enumeration of over 300 explicitly derived invariants. This approach to invariants was subsequently supplanted by the more abstract and generalized methodology advanced by Hilbert. Although favorably received at the time, Noether later characterized her dissertation and subsequent related publications as "crap." Her subsequent research endeavors diverged entirely into a distinct domain.

University of Erlangen–Nuremberg

From 1908 to 1915, Noether served as an unpaid lecturer at Erlangen's Mathematical Institute, periodically deputizing for her father, Max Noether, when he was incapacitated by illness from lecturing. She became a member of the Circolo Matematico di Palermo in 1908 and the Deutsche Mathematiker-Vereinigung in 1909. In 1910 and 1911, she issued publications extending her doctoral research from three variables to n variables.

Gordan retired in 1910, and Noether continued her instructional duties under the guidance of his successors, Erhard Schmidt and Ernst Fischer, who assumed the position from the former in 1911. According to her colleague Hermann Weyl and her biographer Auguste Dick, Fischer exerted a significant influence on Noether, notably by familiarizing her with the contributions of David Hilbert. Noether and Fischer cultivated a vibrant intellectual rapport regarding mathematics and frequently engaged in extensive post-lecture discussions; Noether is reportedly dispatched postcards to Fischer, thereby extending her mathematical deliberations.

Between 1913 and 1916, Noether authored multiple publications that expanded upon and applied Hilbert's methodologies to mathematical constructs, including fields of rational functions and the invariants of finite groups. This period represented Noether's initial engagement with abstract algebra, a domain where she would subsequently achieve seminal advancements.

While in Erlangen, Noether provided guidance to two doctoral candidates, Hans Falckenberg and Fritz Seidelmann, who successfully defended their dissertations in 1911 and 1916, respectively. Notwithstanding Noether's substantial involvement, both students were formally supervised by her father. Subsequent to earning his doctorate, Falckenberg held positions in Braunschweig and Königsberg prior to his appointment as a professor at the University of Giessen, whereas Seidelmann attained a professorship in Munich.

The University of Göttingen

Habilitation and the Development of Noether's Theorem

In early 1915, David Hilbert and Felix Klein extended an invitation to Noether to rejoin the University of Göttingen. Their endeavor to appoint her encountered initial resistance from philologists and historians within the philosophical faculty, who maintained that women were unsuitable for the position of privatdozenten. During a departmental meeting convened to discuss the issue, a faculty member voiced opposition, stating: "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?" Hilbert, asserting that Noether's qualifications were the sole pertinent factor and that the candidate's gender was immaterial, vehemently objected and reprimanded those opposing her habilitation. Although his precise words are not extant, his objection is frequently reported to have included the assertion that the university was "not a bathhouse". Pavel Alexandrov's recollections indicate that faculty opposition to Noether stemmed not only from sexism but also from disapproval of her social-democratic political convictions and Jewish heritage.

Noether relocated to Göttingen in late April; a fortnight thereafter, her mother unexpectedly passed away in Erlangen. While she had previously undergone medical treatment for an ocular condition, its specific nature and influence on her demise remain undetermined. Concurrently, Noether's father retired, and her brother enlisted in the German Army for service in World War I. She subsequently returned to Erlangen for a period of several weeks, primarily to attend to her elderly father.

During her initial years of instruction at Göttingen, she held no official appointment and received no remuneration. Her lectures were frequently publicized under Hilbert's name, with Noether providing "assistance".

Shortly after her arrival at Göttingen, she demonstrated her intellectual prowess by formulating what is now recognized as Noether's theorem, which establishes a fundamental connection between conservation laws and differentiable symmetries within a physical system. Her seminal paper, titled Invariante Variationsprobleme, was presented by her colleague, Felix Klein, on July 26, 1918, during a session of the Royal Society of Sciences at Göttingen. Noether presumably did not present the work personally, owing to her non-membership in the society. In their publication Symmetry and the Beautiful Universe, American physicists Leon M. Lederman and Christopher T. Hill contend that Noether's theorem stands as "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".

The conclusion of World War I and the subsequent German Revolution of 1918–1919 precipitated substantial shifts in societal norms, encompassing an expansion of women's rights. Consequently, in 1919, the University of Göttingen authorized Noether to pursue her habilitation, a prerequisite for tenure. Her oral examination took place in late May, followed by the successful delivery of her habilitation lecture in June 1919. Noether subsequently attained the status of privatdozent, and during the ensuing fall semester, she presented the inaugural lectures officially attributed to her. Despite these advancements, she continued to receive no compensation for her academic contributions.

Three years subsequent to this, Otto Boelitz, the Prussian Minister for Science, Art, and Public Education, formally bestowed upon her the title of nicht beamteter ausserordentlicher Professor, signifying an untenured professor with restricted internal administrative responsibilities. This designation represented an unpaid "extraordinary" professorship, distinct from the more senior "ordinary" professorship, which constituted a civil-service appointment. While acknowledging the significance of her contributions, this role did not include a salary. Noether's lectures remained unremunerated until her appointment to the specialized role of Lehrbeauftragte für Algebra (Lecturer for Algebra) the following year.

Contributions to Abstract Algebra

Noether's theorem profoundly influenced classical and quantum mechanics; however, within the mathematical community, she is primarily recognized for her seminal contributions to abstract algebra. Nathan Jacobson, in his introduction to Noether's Collected Papers, articulated that:

The development of abstract algebra, a singularly distinctive innovation in twentieth-century mathematics, is largely attributable to her contributions, evident in her published papers, lectures, and personal influence on her contemporaries.

Noether initiated her algebraic research in 1920, co-authoring a paper with her protégé Werner Schmeidler. This publication focused on the theory of ideals, wherein they established definitions for left and right ideals within a ring structure.

The subsequent year, she published Idealtheorie in Ringbereichen, a paper that analyzed ascending chain conditions concerning mathematical ideals. In this work, she provided a comprehensive proof of the Lasker–Noether theorem. Prominent algebraist Irving Kaplansky characterized this contribution as "revolutionary." This publication also led to the coinage of the term Noetherian to describe mathematical objects that fulfill the ascending chain condition.

In 1924, Bartel Leendert van der Waerden, a young Dutch mathematician, commenced his studies at the University of Göttingen. He promptly collaborated with Noether, who furnished him with indispensable methodologies for abstract conceptualization. Van der Waerden subsequently remarked that her originality was "absolute beyond comparison." Upon his return to Amsterdam, he authored Moderne Algebra, a foundational two-volume treatise in the field. The second volume, released in 1931, drew extensively from Noether's research. Although Noether did not actively pursue recognition, van der Waerden acknowledged her contributions in a note within the seventh edition, stating the work was "based in part on lectures by E. Artin and E. Noether." From 1927 onward, Noether engaged in close collaboration with Emil Artin, Richard Brauer, and Helmut Hasse on the subject of noncommutative algebras.

Van der Waerden's presence at Göttingen coincided with a broader influx of mathematicians globally, as the university had evolved into a preeminent center for mathematical and physical inquiry. Russian mathematicians Pavel Alexandrov and Pavel Urysohn were among the initial international visitors in 1923. From 1926 to 1930, Alexandrov delivered regular lectures at the university, fostering a close friendship with Noether. He affectionately referred to her as der Noether, employing der as an honorific rather than its conventional masculine German article usage. Noether endeavored to facilitate his appointment as a regular professor at Göttingen, but ultimately succeeded only in assisting him to secure a Rockefeller Foundation scholarship for the 1927–1928 academic year at Princeton University.

Doctoral Students

At Göttingen, Noether oversaw the doctoral studies of over twelve students; however, due to institutional restrictions preventing her from independently supervising dissertations, most were co-supervised with Edmund Landau and other faculty members. Her inaugural doctoral student was Grete Hermann, who successfully defended her dissertation in February 1925. While Hermann is primarily recognized for her contributions to the foundations of quantum mechanics, her dissertation itself was regarded as a significant advancement in ideal theory. Hermann subsequently referred to Noether with reverence as her "dissertation-mother."

Concurrently, Heinrich Grell and Rudolf Hölzer completed their dissertations under Noether's guidance. Tragically, Hölzer succumbed to tuberculosis shortly before his scheduled defense. Grell successfully defended his thesis in 1926 and subsequently held positions at the University of Jena and the University of Halle. In 1935, he lost his teaching license following accusations of homosexual acts, but was later reinstated, ultimately becoming a professor at Humboldt University in 1948.

Emmy Noether subsequently advised Werner Weber and Jakob Levitzki, both of whom successfully defended their doctoral theses in 1929. Weber, despite being regarded as a mathematician of limited distinction, later participated in the expulsion of Jewish mathematicians from Göttingen. Levitzki, conversely, held positions at Yale University before joining the Hebrew University of Jerusalem in British-ruled Mandatory Palestine, where he made substantial contributions to ring theory, notably through Levitzky's theorem and the Hopkins–Levitzki theorem.

Additional students mentored by Noether, often referred to as "Noether Boys," included Max Deuring, Hans Fitting, Ernst Witt, Chiungtze C. Tsen, and Otto Schilling. Deuring, widely considered Noether's most promising student, earned his doctorate in 1930. His career involved work in Hamburg, Marden, and Göttingen, where he became recognized for his significant contributions to arithmetic geometry. Fitting completed his graduation in 1931 with a thesis focused on abelian groups and is remembered for his foundational work in group theory, specifically Fitting's theorem and the Fitting lemma. Tragically, he passed away at 31 due to a bone disease.

Ernst Witt initially pursued his studies under Noether's guidance; however, her academic position was rescinded in April 1933, leading to his reassignment to Gustav Herglotz. Witt obtained his PhD in July 1933, submitting a thesis on the Riemann-Roch theorem and zeta-functions, and subsequently made several notable contributions that are now eponymously associated with him. Chiungtze C. Tsen, primarily recognized for establishing Tsen's theorem, received his doctorate in December of the same year. He returned to China in 1935, commencing his teaching career at National Chekiang University, but passed away just five years later. Otto Schilling also began his doctoral studies with Noether but was compelled to seek a new supervisor following her emigration. He completed his PhD in 1934 at the University of Marburg under the advisement of Helmut Hasse. Subsequently, he undertook postdoctoral research at Trinity College, Cambridge, prior to relocating to the United States.

Among Noether's other doctoral students were Wilhelm Dörnte, who earned his doctorate in 1927 with a thesis on groups; Werner Vorbeck, who completed his doctorate in 1935 with a thesis on splitting fields; and Wolfgang Wichmann, whose doctorate in 1936 focused on p-adic theory. While details regarding Dörnte and Vorbeck remain unavailable, it is documented that Wichmann actively supported a student initiative that unsuccessfully sought to overturn Noether's dismissal. He subsequently died as a soldier on the Eastern Front during World War II.

The Noether School

Beyond her direct doctoral students, Noether cultivated a close community of mathematicians who embraced her methodology in abstract algebra and significantly advanced the field's development; this collective is frequently termed the "Noether school." A notable instance of this collaboration is her extensive work with Wolfgang Krull, whose contributions, including his Hauptidealsatz and dimension theory for commutative rings, substantially propelled commutative algebra. Similarly, Gottfried Köthe advanced the theory of hypercomplex quantities by applying methods developed by Noether and Krull.

Beyond her profound mathematical acumen, Noether was esteemed for her interpersonal consideration. Although she occasionally exhibited brusqueness toward dissenting colleagues, she cultivated a reputation for helpfulness and patient mentorship of nascent students. Her unwavering commitment to mathematical precision prompted one colleague to characterize her as "a severe critic," yet she harmonized this rigorous demand for accuracy with a supportive and nurturing demeanor. In Noether's obituary, Van der Waerden offered the following description:

Entirely devoid of ego and vanity, she never sought personal recognition, but rather prioritized and championed the achievements of her students above all else.

Noether demonstrated an exceptional dedication to both her discipline and her students, extending well beyond conventional academic hours. On one occasion, when the university building was inaccessible due to a state holiday, she convened her class on the exterior steps, guided them through a wooded area, and delivered her lecture at a nearby coffee house. Subsequently, following her dismissal from teaching by Nazi Germany, she extended invitations to students to her residence, where they engaged in discussions concerning their future plans and various mathematical concepts.

Impactful Lectures

Initially, Noether's austere lifestyle stemmed from the university's refusal to compensate her for her academic contributions. Even after the university commenced paying her a modest salary in 1923, she maintained a simple and unostentatious existence. Although her remuneration increased later in her life, she consistently saved half of her earnings with the intention of bequeathing them to her nephew, Gottfried E. Noether.

Biographers indicate that Emmy Noether prioritized her academic pursuits over concerns about personal appearance and social etiquette. Olga Taussky-Todd, a prominent algebraist who studied under Noether, recounted an instance at a luncheon where Noether, deeply absorbed in a mathematical discussion, "gesticulated wildly" while eating, "spilled her food constantly," and "wiped it off from her dress, completely unperturbed." Students attentive to decorum were reportedly discomfited by her retrieving a handkerchief from her blouse and her disregard for her increasingly disheveled hair during lectures. On one occasion, two female students attempted to convey their concerns during a break in a two-hour class, but they found themselves unable to interrupt her animated mathematical discourse with other students.

Noether's lectures were not structured by a formal lesson plan. Her rapid delivery made her presentations challenging to comprehend for many, including notable mathematicians Carl Ludwig Siegel and Paul Dubreil. Students who found her pedagogical approach uncongenial frequently experienced a sense of detachment. Visiting "outsiders" attending Noether's lectures often departed within thirty minutes, citing frustration or confusion. A regular student once remarked on such an occurrence, stating, "The enemy has been defeated; he has cleared out."

Noether utilized her lectures as an interactive forum for spontaneous discussions with her students, facilitating the exploration and elucidation of significant mathematical problems. Several of her most crucial findings emerged from these lecture sessions, and the notes compiled by her students subsequently served as foundational material for influential textbooks, including those authored by van der Waerden and Deuring. She instilled a contagious mathematical fervor in her most committed students, who highly valued their dynamic intellectual exchanges with her.

Many of Noether's colleagues attended her lectures, and she occasionally permitted others, including her students, to receive attribution for her concepts, leading to a substantial portion of her contributions appearing in publications not bearing her name. Records indicate that Noether delivered a minimum of five semester-long courses at Göttingen:

• Winter 1924–1925: Gruppentheorie und hyperkomplexe Zahlen [Group Theory and Hypercomplex Numbers]
• Winter 1927–1928: Hyperkomplexe Grössen und Darstellungstheorie [Hypercomplex Quantities and Representation Theory]
• Summer 1928: Nichtkommutative Algebra [Noncommutative Algebra]
• Summer 1929: Nichtkommutative Arithmetik [Noncommutative Arithmetic]
• Winter 1929–1930: Algebra der hyperkomplexen Grössen [Algebra of Hypercomplex Quantities]

Moscow State University

During the academic year 1928–1929, Noether accepted an invitation to Moscow State University, where she resumed her collaboration with P. S. Alexandrov. Beyond her ongoing research, she delivered courses in abstract algebra and algebraic geometry. She also engaged with the distinguished topologists Lev Pontryagin and Nikolai Chebotaryov, both of whom subsequently lauded her significant contributions to the advancement of Galois theory.

While politics was not the primary focus of her life, Noether demonstrated a strong interest in political affairs and, as noted by Alexandrov, expressed substantial support for the Russian Revolution. She particularly welcomed Soviet advancements in science and mathematics, viewing them as evidence of new possibilities fostered by the Bolshevik initiative. This perspective led to difficulties for her in Germany, culminating in her expulsion from a pension lodging after student leaders lodged complaints about residing with "a Marxist-leaning Jewess." Hermann Weyl recounted that "During the wild times after the Revolution of 1918," Noether "sided more or less with the Social Democrats." She was affiliated with the Independent Social Democrats, a short-lived splinter party, from 1919 to 1922. Logician and historian Colin McLarty characterized her stance by stating, "she was not a Bolshevist, but was not afraid to be called one."

Noether intended to return to Moscow, an endeavor supported by Alexandrov. Following her departure from Germany in 1933, Alexandrov attempted to facilitate her appointment to a professorial chair at Moscow State University via the Soviet Education Ministry. Although this effort was unsuccessful, they maintained frequent correspondence throughout the 1930s, and by 1935, she had formulated plans for a return to the Soviet Union.

Recognition

In 1932, Emmy Noether and Emil Artin were honored with the Ackermann–Teubner Memorial Award for their significant mathematical contributions. The award, which included a monetary prize of 500 ℛ︁ℳ︁, was widely regarded as a belated official acknowledgment of her substantial achievements in the discipline. Despite this recognition, her peers voiced dissatisfaction that she had not been elected to the Göttingen Gesellschaft der Wissenschaften (academy of sciences) and had never attained the rank of Ordentlicher Professor (full professor).

In 1932, Noether's fiftieth birthday was commemorated by her colleagues in a manner characteristic of mathematicians. Helmut Hasse dedicated an article to her in the Mathematische Annalen, where he substantiated her hypothesis that certain facets of noncommutative algebra are less complex than their commutative counterparts, through the demonstration of a noncommutative reciprocity law. This discovery brought her considerable satisfaction. Additionally, Hasse presented her with a mathematical enigma, termed the "m_μν-riddle of syllables," which she promptly resolved; however, the riddle itself is no longer extant.

In September of the same year, Noether presented a plenary address (großer Vortrag) titled "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich. The congress attracted 800 attendees, among whom were her colleagues Hermann Weyl, Edmund Landau, and Wolfgang Krull. The event featured 420 official participants and twenty-one plenary presentations. Noether's distinguished speaking slot seemingly underscored the significance of her mathematical contributions. The 1932 congress is occasionally characterized as the zenith of her professional trajectory.

Dismissal from Göttingen by Nazi Germany

Following Adolf Hitler's appointment as German Reichskanzler in January 1933, Nazi activities intensified significantly across the nation. At the University of Göttingen, the German Student Association spearheaded a campaign against the "un-German spirit" associated with Jewish individuals, receiving support from privatdozent and Noether's former student, Werner Weber. This pervasive antisemitism fostered an environment overtly hostile towards Jewish professors. A young demonstrator was reportedly quoted as stating: "Aryan students demand Aryan mathematics, not Jewish mathematics."

Among the initial legislative measures enacted by Hitler's administration was the Law for the Restoration of the Professional Civil Service. This legislation mandated the dismissal of Jewish individuals and politically suspect government employees, including university professors, from their positions, unless they could prove their "loyalty to Germany" through service in World War I. In April 1933, Noether received an official notification from the Prussian Ministry for Sciences, Art, and Public Education, which stated: "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." Concurrently, several of Noether's colleagues, such as Max Born and Richard Courant, also experienced the revocation of their appointments.

Noether responded to the decision with composure, offering assistance to others amidst the prevailing adversity. Hermann Weyl subsequently remarked 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." Characteristically, Noether maintained her focus on mathematical pursuits, convening students in her residence to deliberate on class field theory. Upon the appearance of one of her students in the uniform of the Nazi paramilitary organization Sturmabteilung (SA), she exhibited no signs of distress and, according to reports, even found humor in the situation later.

Seeking Refuge at Bryn Mawr and Princeton

As numerous recently unemployed professors sought employment beyond Germany's borders, their counterparts in the United States endeavored to offer support and professional opportunities. Albert Einstein and Hermann Weyl secured appointments at the Institute for Advanced Study in Princeton, while other academics worked to identify sponsors essential for legal immigration. Noether received overtures from representatives of two academic institutions: Bryn Mawr College in the United States and Somerville College at the University of Oxford in England. Following extensive discussions with the Rockefeller Foundation, a grant was approved for Noether to join Bryn Mawr, where she commenced her new role in late 1933.

During her tenure at Bryn Mawr, Noether established a friendship with Anna Wheeler, who had previously pursued studies at Göttingen prior to Noether's arrival. Further institutional support was provided by Bryn Mawr's president, Marion Edwards Park, who actively encouraged local mathematicians to observe Dr. Noether's work.

While at Bryn Mawr, Noether cultivated a research group, informally known as the 'Noether girls,' comprising four postdoctoral researchers—Grace Shover Quinn, Marie Johanna Weiss, and Olga Taussky-Todd, all of whom subsequently achieved distinguished careers in mathematics—and one doctoral student, Ruth Stauffer. This group diligently engaged with van der Waerden's Moderne Algebra I and selections from Erich Hecke's Theorie der algebraischen Zahlen (Theory of algebraic numbers). Ruth Stauffer was Noether's sole doctoral candidate in the United States; however, Noether passed away shortly before Stauffer's graduation. Stauffer successfully completed her doctoral examination with Richard Brauer, earning her degree in June 1935 with a dissertation on separable normal extensions. Following her doctorate, Stauffer pursued a brief career in teaching before dedicating over three decades to work as a statistician.

In 1934, Noether commenced lecturing at the Institute for Advanced Study in Princeton, following an invitation extended by Abraham Flexner and Oswald Veblen. During this period, she collaborated with Abraham Albert and Harry Vandiver. Regarding Princeton University, she notably commented on her perceived unwelcome status at "the men's university, where nothing female is admitted."

Noether's tenure in the United States proved agreeable, characterized by a supportive academic environment and deep engagement with her primary research interests. In mid-1934, she made a brief Fritz Noether, having been dismissed from his position at the Technische Hochschule Breslau, had subsequently accepted an appointment at the Research Institute for Mathematics and Mechanics in Tomsk, located within the Siberian Federal District of Russia.

Although numerous former colleagues had been displaced from their university positions, Noether was permitted to utilize the Göttingen library facilities as a "foreign scholar." Subsequently, she returned to the United States without incident, resuming her academic pursuits at Bryn Mawr.

Death

In April 1935, medical professionals identified a tumor in Noether's pelvis. Concerns regarding potential surgical complications led to a preliminary two-day period of bed rest. During the subsequent operation, an ovarian cyst, described as "the size of a large cantaloupe," was discovered. Two smaller uterine tumors appeared benign and were not excised to prevent extending the surgical duration. For three days post-operation, Noether exhibited normal convalescence, and she rapidly recovered from a circulatory collapse on the fourth day. However, on April 14, Noether lost consciousness, her temperature escalated to 109 °F (42.8 °C), and she succumbed. One attending physician noted, "[I]t is not easy to say what had occurred in Dr. Noether," postulating, "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 years old at the time of her passing.

Days following Noether's death, a private memorial service was conducted by her friends and colleagues at Bryn Mawr, hosted at the residence of College President Park. Hermann Weyl and Richard Brauer journeyed from Princeton to deliver eulogies. In the ensuing months, numerous written tributes emerged internationally, with notable figures such as Albert Einstein, van der Waerden, Weyl, and Pavel Alexandrov offering their respects. Her remains were cremated, and the ashes were interred beneath the walkway encircling the cloisters of the Old Library at Bryn Mawr.

Contributions to Mathematics and Physics

Noether's contributions to abstract algebra and topology significantly influenced the field of mathematics; concurrently, Noether's theorem holds extensive implications for theoretical physics and dynamical systems. She demonstrated a profound aptitude for abstract conceptualization, enabling her to formulate novel and innovative approaches to mathematical problems. Her esteemed colleague and friend, Hermann Weyl, categorized her scholarly achievements into three distinct periods:

(1) The period of relative dependence, spanning 1907–1919.

(2) Investigations centered on the general theory of ideals, conducted from 1920–1926.

(3) The examination of non-commutative algebras, their representations through linear transformations, and their subsequent application to the analysis of commutative number fields and their associated arithmetics.

During her first epoch (1907–1919), Noether primarily addressed differential and algebraic invariants, commencing with her doctoral research under Paul Gordan. Her mathematical scope expanded, and her work evolved towards greater generality and abstraction, through her engagement with David Hilbert's contributions and collaborative exchanges with Gordan's successor, Ernst Sigismund Fischer. Soon after relocating to Göttingen in 1915, she established Noether's two theorems, recognized as "one of the most important mathematical theorems ever proved in guiding the development of modern physics".

In her second epoch (1920–1926), Noether dedicated her efforts to advancing the theory of mathematical rings. Subsequently, in the third epoch (1927–1935), she concentrated on noncommutative algebra, linear transformations, and commutative number fields. While the results from Noether's first epoch were noteworthy and valuable, her prominence among mathematicians is primarily attributed to the pioneering contributions made during her second and third epochs, as highlighted in her obituaries by Hermann Weyl and B. L. van der Waerden.

Across these epochs, she did not simply apply existing ideas and methodologies from earlier mathematicians; instead, she formulated novel systems of mathematical definitions that subsequently influenced future mathematical endeavors. Specifically, she established an entirely new theory of ideals in rings, extending the foundational work of Richard Dedekind. Furthermore, she is recognized for introducing ascending chain conditions – a straightforward finiteness criterion that proved remarkably effective in her applications. These conditions, coupled with the theory of ideals, allowed Noether to generalize numerous prior findings and approach established problems from a novel viewpoint, including algebraic invariants, a subject previously explored by her father, and elimination theory.

Noether's paramount contributions to mathematics involved the advancement of the nascent field of abstract algebra.

Distinguishing her from many contemporaries, Noether's approach to abstraction did not involve generalization from specific examples; instead, she engaged directly with abstract concepts. As recounted by van der Waerden in her obituary,

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."

This approach exemplifies begriffliche Mathematik (purely conceptual mathematics), a hallmark of Noether's methodology. Subsequently, this mathematical style gained adoption among other mathematicians, particularly within the emerging domain of abstract algebra.

First Epoch (1908–1919)

Algebraic Invariant Theory

A significant portion of Noether's early career, during her first epoch, focused on invariant theory, particularly algebraic invariant theory. Invariant theory investigates mathematical expressions that retain their value (i.e., remain invariant) under specific groups of transformations. For instance, in a common physical analogy, rotating a rigid meter stick alters the coordinates of its endpoints, yet its length persists unchanged. A more complex illustration of an invariant is the discriminant B§56§ − 4AC of a homogeneous quadratic polynomial Ax§1314§ + Bxy + Cy§1920§, where x and y represent indeterminates. This discriminant is termed "invariant" due to its constancy under linear substitutions x → ax + by and y → cx + dy, provided their determinant ad − bc equals 1. Collectively, these substitutions constitute the special linear group SL§5152§.

The inquiry can extend to identifying all polynomials in A, B, and C that remain invariant under the action of SL§910§; these are, in fact, polynomials of the discriminant. More broadly, one may seek the invariants of homogeneous polynomials of higher degree, such as A§1516§x^ry§2526§ + ... + A_rx§3132§y^r, which manifest as specific polynomials in the coefficients A§4344§, ..., A_r. This line of questioning can be further extended to homogeneous polynomials involving more than two variables.

A primary objective of invariant theory involved resolving the "finite basis problem." This problem investigated whether all invariants could be derived from a finite set of initial invariants, termed generators, through iterative addition or multiplication, given that the sum or product of any two invariants also constitutes an invariant. For instance, the discriminant provides a finite basis, comprising a single element, for the invariants of a quadratic polynomial.

Paul Gordan, Noether's academic advisor, earned renown as the "king of invariant theory," with his seminal mathematical contribution being the 1870 resolution of the finite basis problem for invariants of homogeneous polynomials in two variables. Gordan's proof presented a constructive methodology for identifying all invariants and their respective generators; however, he could not extend this approach to invariants involving three or more variables. Subsequently, in 1890, David Hilbert established an analogous theorem for the invariants of homogeneous polynomials across an arbitrary number of variables. Notably, Hilbert's methodology applied not only to the special linear group but also to various of its subgroups, including the special orthogonal group.

Emulating Gordan's scholarly trajectory, Noether dedicated her doctoral dissertation and several subsequent publications to invariant theory. Her work expanded upon Gordan's findings and integrated Hilbert's research. Nevertheless, she later expressed disdain for this early work, deeming it of minor significance and confessing to having forgotten its specific intricacies. Hermann Weyl observed:

[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.

Galois Theory

Galois theory investigates transformations within number fields that reorder the roots of an equation. Consider a polynomial equation involving a variable x of degree n, where its coefficients originate from a specified ground field, such as the field of real numbers, rational numbers, or integers modulo 7. Solutions for x that cause this polynomial to evaluate to zero are termed roots, though such solutions may not always exist within the initial field. For instance, if the polynomial is x§1516§ + 1 and the ground field is the real numbers, no roots exist, as any real value for x results in the polynomial being greater than or equal to one. However, extending the field can introduce roots, and a sufficiently extended field will invariably contain a number of roots equivalent to the polynomial's degree.

Extending the preceding illustration, if the field is expanded to encompass complex numbers, the polynomial acquires two roots: +i and −i, where i represents the imaginary unit, defined by i ² = −1. Broadly, the extension field within which a polynomial can be completely factored into its constituent roots is designated as the splitting field of that polynomial.

The Galois group of a polynomial is defined as the collection of all transformations of its splitting field that maintain both the ground field and the polynomial's roots. (These transformations are specifically termed automorphisms.) For the polynomial x§45§ + 1, its Galois group comprises two elements: the identity transformation, which maps every complex number to itself, and complex conjugation, which transforms +i to −i. As the Galois group preserves the ground field, it consequently leaves the polynomial's coefficients unaltered and, by extension, the entire set of roots. Each root may be mapped to another root, implying that each transformation establishes a permutation among the n roots. The profound importance of the Galois group stems from the fundamental theorem of Galois theory, which demonstrates a one-to-one correspondence between the intermediate fields situated between the ground field and the splitting field, and the subgroups of the Galois group.

Noether's 1918 publication addressed the inverse Galois problem. Rather than focusing on identifying the Galois group of transformations for a specified field and its extension, Noether investigated whether an extension of a given field could invariably be found to possess a particular group as its Galois group. This inquiry was subsequently reduced to "Noether's problem," which questions whether the fixed field of a subgroup G within the permutation group S_n, when acting on the field k(x§1516§, ..., x_n), consistently constitutes a pure transcendental extension of the field k. Noether initially presented this problem in a 1913 paper, crediting its origin to her colleague Fischer. She demonstrated its validity for cases where n equals 2, 3, or 4. However, in 1969, Richard Swan identified a counterexample to Noether's problem, specifically involving n = 47 and G as a cyclic group of order 47 (despite this particular group being realizable as a Galois group over the rationals through alternative constructions). The inverse Galois problem continues to be an unresolved mathematical challenge.

Physics

In 1915, David Hilbert and Felix Klein invited Noether to Göttingen, seeking her specialized knowledge in invariant theory to aid their comprehension of general relativity, a geometric theory of gravitation primarily developed by Albert Einstein. Hilbert had noted an apparent violation of energy conservation within general relativity, attributing this to the capacity of gravitational energy to exert its own gravitational influence. Noether resolved this paradox and introduced a foundational instrument for modern theoretical physics in a 1918 publication. This seminal paper introduced two theorems, the first of which is universally recognized as Noether's theorem. Collectively, these theorems not only addressed the issue within general relativity but also established the conserved quantities for every physical system characterized by continuous symmetry. Following his review of her work, Einstein communicated to Hilbert:

I received a highly engaging paper on invariants from Miss Noether yesterday. I am impressed by the capacity to comprehend such concepts with such generality. The established academics at Göttingen ought to learn from Miss Noether; her expertise appears profound.

For instance, if a physical system exhibits identical behavior irrespective of its spatial orientation, its governing physical laws are considered rotationally symmetric; Noether's theorem demonstrates that this symmetry necessitates the conservation of the system's angular momentum. The physical system itself does not require inherent symmetry; for example, a jagged asteroid rotating in space still conserves angular momentum despite its irregular form. Instead, the conservation law arises from the symmetry inherent in the physical laws that govern the system. Furthermore, if a physical experiment yields consistent results regardless of its location or time, its underlying laws possess symmetry under continuous spatial and temporal translations; Noether's theorem establishes that these symmetries correspond to the conservation laws of linear momentum and energy, respectively, within that system.

Contemporaneously, physicists lacked familiarity with Sophus Lie's theory of continuous groups, which formed the foundational basis for Noether's work. A significant number of physicists initially encountered Noether's theorem through an article by Edward Lee Hill, which, however, presented only a specialized instance of the theorem. As a result, the comprehensive implications of her findings were not immediately recognized. Nevertheless, in the latter half of the 20th century, Noether's theorem evolved into a cornerstone of modern theoretical physics, valued for both its profound insights into conservation laws and its utility as a practical computational instrument. This theorem enables researchers to deduce conserved quantities directly from the observed symmetries inherent in a physical system. Conversely, it aids in characterizing a physical system by referencing categories of hypothetical physical laws. To illustrate, consider the hypothetical discovery of a novel physical phenomenon. Noether's theorem offers a crucial test for theoretical models explaining such a phenomenon: if a theory incorporates a continuous symmetry, the theorem guarantees the existence of a conserved quantity, and for the theory to be valid, this conservation must be empirically verifiable through experimentation.

Second Epoch (1920–1926)

Ascending and Descending Chain Conditions

During this period, Noether gained recognition for her skillful application of ascending (Teilerkettensatz) and descending (Vielfachenkettensatz) chain conditions. An ascending sequence of non-empty subsets, such as A§78§, A§1112§, A§1516§, ..., within a set S is conventionally defined by each subset being contained within the subsequent one.

A §1011§ ⊆ A §2122§ ⊆ A §3233§ ⊆ ⋯ . {\displaystyle A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \cdots .}

Conversely, a sequence of subsets within S is termed descending when each successive subset is contained within its predecessor.

A §1011§ ⊇ A §2122§ ⊇ A §3233§ ⊇ ⋯ . {\displaystyle A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \cdots .}

A chain is defined as becoming constant after a finite number of steps if an integer n exists such that ${\displaystyle A_{n}=A_{m}}$ for all m ≥ n. The ascending chain condition is satisfied by a collection of subsets within a given set if every ascending sequence eventually stabilizes. Similarly, the descending chain condition is met if any descending sequence also stabilizes after a finite number of steps. These chain conditions are instrumental in demonstrating the existence of maximal or minimal elements within any set of sub-objects, or in proving that intricate objects can be generated from a reduced number of constituent elements.

Numerous algebraic structures in abstract algebra can fulfill chain conditions; typically, those satisfying an ascending chain condition are designated as Noetherian, a tribute to her contributions. Specifically, a Noetherian ring is characterized by satisfying an ascending chain condition on both its left and right ideals. In contrast, a Noetherian group is defined as one where every strictly ascending chain of subgroups is finite. A Noetherian module is a module in which every strictly ascending chain of submodules stabilizes after a finite number of steps. Furthermore, a Noetherian space refers to a topological space whose open subsets adhere to the ascending chain condition, thereby classifying the spectrum of a Noetherian ring as a Noetherian topological space.

The chain condition frequently exhibits an inheritance property among sub-objects. For instance, all subspaces within a Noetherian space are themselves Noetherian; similarly, all subgroups and quotient groups derived from a Noetherian group are also Noetherian. Analogously, mutatis mutandis, this principle extends to submodules and quotient modules of a Noetherian module. Moreover, the chain condition can be inherited by various combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings retain the Noetherian property, as does the ring of formal power series constructed over a Noetherian ring.

Noetherian induction, also termed well-founded induction, represents a further application of these chain conditions and serves as a generalization of mathematical induction. This method is frequently employed to simplify general assertions concerning object collections into statements about particular objects within those collections. Consider S as a partially ordered set. A common approach to establish a statement about elements within S involves positing the existence of a counterexample and subsequently deriving a contradiction, thus demonstrating the contrapositive of the initial assertion. The fundamental principle of Noetherian induction asserts that every non-empty subset of S must contain a minimal element. Specifically, the collection of all counterexamples will include a minimal element, referred to as the minimal counterexample. Consequently, to validate the original statement, it is sufficient to demonstrate a seemingly less stringent condition: that for any given counterexample, a smaller counterexample exists.

Commutative Rings, Ideals, and Modules

Noether's seminal 1921 publication, titled Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains), established the groundwork for general commutative ring theory and presented one of the earliest comprehensive definitions of a commutative ring. Prior to her work, the majority of findings in commutative algebra were confined to specific instances of commutative rings, including polynomial rings over fields or rings of algebraic integers. Noether demonstrated that within any ring satisfying the ascending chain condition on ideals, every ideal is finitely generated. The French mathematician Claude Chevalley introduced the term Noetherian ring in 1943 to characterize this specific property. A significant contribution of Noether's 1921 paper is the Lasker–Noether theorem, which broadens Lasker's original theorem concerning the primary decomposition of ideals in polynomial rings to encompass all Noetherian rings. This theorem can be conceptualized as an extension of the fundamental theorem of arithmetic, which posits that every positive integer possesses a unique factorization into prime numbers.

In her 1927 publication, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields), Noether delineated the characteristics of rings where ideals exhibit unique factorization into prime ideals, now recognized as Dedekind domains. She demonstrated that these rings are defined by five specific criteria: they must adhere to both ascending and descending chain conditions, contain a unit element while lacking zero divisors, and be integrally closed within their corresponding field of fractions. This paper additionally presents what are now known as the isomorphism theorems, which elucidate fundamental natural isomorphisms, alongside other foundational findings concerning Noetherian and Artinian modules.

Elimination Theory

Between 1923 and 1924, Noether extended her ideal theory to elimination theory, employing a formulation she credited to her student, Kurt Hentzelt. Her work demonstrated that core theorems pertaining to polynomial factorization were directly transferable to this context.

Historically, elimination theory has focused on the process of removing one or more variables from a system of polynomial equations, frequently utilizing the method of resultants. For illustrative purposes, a system of equations can often be expressed in the following form:

Mv = 0

In this representation, a matrix (or linear transformation) M, independent of the variable x, multiplied by a vector v (containing only non-zero powers of x), yields the zero vector, §89§. Consequently, the determinant of matrix M must equate to zero, thereby furnishing a novel equation from which the variable x has been successfully eliminated.

Invariant Theory of Finite Groups

Earlier methods, such as Hilbert's non-constructive solution to the finite basis problem, lacked the capacity to provide quantitative data regarding the invariants of a group action and were not universally applicable to all group actions. In her 1915 publication, Noether presented a solution to the finite basis problem for a finite group of transformations G operating on a finite-dimensional vector space over a field with characteristic zero. Her findings demonstrated that the ring of invariants is generated by homogeneous invariants whose degree does not exceed the order of the finite group, a principle known as Noether's bound. Her paper provided two proofs for Noether's bound, both of which are also valid when the field's characteristic is coprime to ${\displaystyle \left|G\right|!}$ (the factorial of the group's order ${\displaystyle \left|G\right|}$). However, the degrees of generators may not adhere to Noether's bound if the field's characteristic divides the number ${\displaystyle \left|G\right|}$. Noether was unable to ascertain the bound's validity when the field's characteristic divides ${\displaystyle \left|G\right|!}$ but not ${\displaystyle \left|G\right|}$. This specific scenario, known as "Noether's gap," remained an unresolved problem for many years until it was independently resolved by Fleischmann in 2000 and Fogarty in 2001, both demonstrating the bound's continued validity.

Noether's 1926 publication expanded Hilbert's theorem to encompass representations of finite groups across any field, particularly addressing the novel scenario where the field's characteristic divides the group's order, a case not covered by Hilbert's original work. William Haboush subsequently broadened Noether's findings to include all reductive groups through his proof of the Mumford conjecture. Within this same paper, Noether also presented the Noether normalization lemma, which establishes that a finitely generated domain A over a field k contains a set {x§1314§, ..., x_n} of algebraically independent elements, such that A is integral over k[x§3132§, ..., x_n].

Topology

Hermann Weyl, in his obituary for Noether, highlighted her significant contributions to topology, underscoring her intellectual generosity and the transformative impact of her insights across various mathematical disciplines. Topology involves the examination of object properties that persist unchanged despite deformation, such as connectivity. A common humorous illustration states that "a topologist cannot distinguish a donut from a coffee mug," given their continuous deformability into one another.

Noether is recognized for pioneering fundamental concepts that facilitated the evolution of algebraic topology from its predecessor, combinatorial topology, particularly through the introduction of homology groups. Alexandrov recounted that during lectures he and Heinz Hopf delivered in 1926 and 1927, Noether "continually made observations which were often deep and subtle," further elaborating that,

Upon encountering the systematic framework of combinatorial topology,

she promptly recognized the value of directly investigating the groups of algebraic complexes and cycles within a given polyhedron, alongside the subgroup of cycles homologous to zero. Rather than adhering to the conventional definition of Betti numbers, she proposed defining the Betti group as the quotient group formed by the group of all cycles and the subgroup of cycles homologous to zero. While this insight appears self-evident today, it represented a fundamentally novel perspective during the period of 1925–1928.

Noether's proposition for an algebraic approach to topology was swiftly embraced by mathematicians such as Hopf and Alexandrov, becoming a prominent subject of discourse among the Göttingen mathematical community. She noted that her concept of a Betti group simplified the comprehension of the Euler–Poincaré formula, and Hopf's subsequent contributions to this field reflected her influence. Noether herself only briefly referenced her topological insights in a 1926 publication, presenting them as an application of group theory.

Concurrently, this algebraic methodology for topology emerged independently in Austria. During a 1926–1927 course delivered in Vienna, Leopold Vietoris introduced the concept of a homology group, which Walther Mayer subsequently formalized into an axiomatic definition in 1928.

Third Epoch (1927–1935)

Hypercomplex Numbers and Representation Theory

Extensive research on hypercomplex numbers and group representations occurred throughout the nineteenth and early twentieth centuries, yet these efforts largely lacked cohesion. Noether synthesized these prior findings, establishing the inaugural general representation theory for groups and algebras. This singular contribution by Noether is credited with initiating a new era in modern algebra and proving foundational for its subsequent evolution.

In essence, Noether integrated the structure theory of associative algebras and the representation theory of groups into a unified arithmetic theory centered on modules and ideals within rings that satisfy ascending chain conditions.

Noncommutative Algebra

Noether also spearheaded several other advancements in algebra. Collaborating with Emil Artin, Richard Brauer, and Helmut Hasse, she established the theory of central simple algebras.

A collaborative publication by Noether, Hasse, and Brauer addressed division algebras, which are algebraic structures permitting division. They demonstrated two significant theorems: first, a local-global theorem asserting that a finite-dimensional central division algebra over a number field, if it splits locally everywhere, also splits globally (thereby becoming trivial); and from this, they derived their Hauptsatz ("main theorem"):

Every finite-dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension.

These theorems facilitate the classification of all finite-dimensional central division algebras over a specified number field. A later publication by Noether demonstrated, as a particular instance of a broader theorem, that all maximal subfields of a division algebra D constitute splitting fields. This paper additionally presents the Skolem–Noether theorem, which posits that any two embeddings of a field extension k into a finite-dimensional central simple algebra over k are conjugate. The Brauer–Noether theorem provides a characterization of the splitting fields for a central division algebra over a field.

Legacy

Noether's contributions remain pertinent to the advancement of theoretical physics and mathematics, solidifying her status as one of the twentieth century's most significant mathematicians. Throughout her life and continuing to the present day, prominent mathematicians including Pavel Alexandrov, Hermann Weyl, and Jean Dieudonné have acclaimed Noether as the most exceptional woman mathematician in recorded history.

In a letter addressed to The New York Times, Albert Einstein articulated:

In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.

In his obituary, fellow algebraist B. L. van der Waerden lauded her mathematical originality as "absolute beyond comparison," while Hermann Weyl asserted that Noether's contributions "changed the face of [abstract] algebra." Mathematician and historian Jeremy Gray observed that Noether's influence is evident in any abstract algebra textbook, stating that "Mathematicians simply do ring theory her way." Her name has been posthumously attributed to numerous mathematical entities and the asteroid 7001 Noether. In 2019, Time magazine commemorated women of the year since 1920 by creating 89 new covers, selecting Noether for the year 1921.

• Timeline of women in science
• Notes

Notes

References

Sources

Selected works by Emmy Noether

Books

Books

• Phillips, Lee (2024), Einstein's Tutor: The Story of Emmy Noether and the Invention of Modern Physics, PublicAffairs, ISBN 9781541702974Hasse, Helmut; Noether, Emmy (2006), Lemmermeyer, Franz; Roquette, Peter (eds.), Helmut Hasse und Emmy Noether – Die Korrespondenz 1925–1935 [Helmut Hasse and Emmy Noether – Their Correspondence 1925–1935] (PDF), Göttingen University, doi:10.17875/gup2006-49, ISBN 978-3-938616-35-2Articles
  • Angier, Natalie (26 March 2012), "The Mighty Mathematician You've Never Heard Of", The New York Times, retrieved 27 January 2024Blue, Meredith (2001), Galois Theory and Noether's Problem (PDF), 34th Annual Meeting of the Mathematical Association of America, MAA Florida Section, archived from the original (PDF) on 29 May 2008, retrieved 9 June 2018Phillips, Lee (26 May 2015), "The female mathematician who changed the course of physics – but couldn't get a job", Ars Technica, California: Condé Nast, retrieved 27 January 2024"Special Issue on Women in Mathematics" (PDF), Notices of the American Mathematical Society, 38 (7), Providence, RI: American Mathematical Society: 701–773, September 1991, ISSN 0002-9920Shen, Qinna (September 2019), "A Refugee Scholar from Nazi Germany: Emmy Noether and Bryn Mawr College", The Mathematical Intelligencer, 41 (3): 52–65, doi:10.1007/s00283-018-9852-0, S2CID 128009850Online biographies
    • Byers, Nina (16 March 2001), "Emmy Noether", Contributions of 20th Century Women to Physics, UCLA, archived from the original on 12 February 2008Taylor, Mandie (22 February 2023), "Emmy Noether", Biographies of Women Mathematicians, Agnes Scott CollegeChown, Marcus (5 March 2025), "Emmy Noether: the genius who taught Einstein", Prospect
      • Emmy Noether at the Mathematics Genealogy Project
      • Noether's application for admission to the University of Erlangen–Nuremberg and three of her curriculum vitae from the Web site of historian Cordula Tollmien

      Media

      • Photograph of Noether taken by Hanna Kunsch — Bryn Mawr College Library Special Collections
      • Photographs of Noether's colleagues and acquaintances from the Web site of Clark Kimberling