Emmy Noether

Amalie Emmy Noether

Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called the "Noether Boys". In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas; her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra. By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world. The following year, Germany's Nazi government dismissed Jews from university positions, and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania. There, she taught doctoral and post-graduate women including Marie Johanna Weiss, Ruth Stauffer, Grace Shover Quinn, and Olga Taussky-Todd. At the same time, she lectured and performed research at the Institute for Advanced Study in Princeton, New Jersey.^[8]

Noether's mathematical work has been divided into three "

Emmy Noether was born on 23 March 1882. She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from Jewish merchant families.^[12] Her first name was "Amalie", but she began using her middle name at a young age and she invariably used the name "Emmy Noether" in her adult life and her publications.^[a]

In her youth, Noether did not stand out academically although she was known for being clever and friendly. She was near-sighted and talked with a minor lisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at an early age.^[13] She was taught to cook and clean, as were most girls of the time, and took piano lessons. She pursued none of these activities with passion, although she loved to dance.^[14]

She had three younger brothers. The eldest, Alfred Noether, was born in 1883 and was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later.^[15] Fritz Noether was born in 1884, studied in Munich and made contributions to applied mathematics. He was executed in the Soviet Union in 1941.^[16] 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.^[17][18]

Education

Noether showed early proficiency in French and English. In the spring of 1900, she took the examination for teachers of these languages and received an overall score of sehr gut (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the

During the 1903–1904 winter semester, she studied at the University of Göttingen, attending lectures given by astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert.^[24]

In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded.^[25] 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.^[26] Under the supervision of Paul Gordan, she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms), in 1907, graduating summa cum laude later that year.^[27] 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.^[28][29] Although it had been well received, Noether later described her thesis and some subsequent similar papers she produced as "crap".^[29][30][b]

University of Erlangen

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.^[31] In 1910 and 1911 she published an extension of her thesis work from three variables to n variables.^[32]

Gordan retired in 1910,^[33] and Noether taught under his successors, Erhard Schmidt and Ernst Fischer, who took over from the former in 1911.^[34] According to her colleague Hermann Weyl and her biographer Auguste Dick, Fischer was an important influence on Noether, in particular by introducing her to the work of David Hilbert.^[35][36] 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. ^[37][38]

From 1913 to 1916 Noether published several papers extending and applying Hilbert's methods to mathematical objects such as fields of rational functions and the invariants of finite groups.^[39] This phase marked Noether's first exposure to abstract algebra, the field to which she would make groundbreaking contributions.^[40]

In Erlangen, Noether advised two doctoral students: Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916. Despite Noether's significant role, they were both officially under the supervision of her father. Following the completion of his doctorate, Falckenberg spent time in Braunschweig and Königsberg before becoming a professor at the University of Giessen while Seidelmann became a professor in Munich.^[41]

University of Göttingen

Habilitation and Noether's theorem

In the spring of 1915, Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein. Their effort to recruit her was initially blocked by the philologists and historians among the philosophical faculty, who insisted that women should not become privatdozenten. 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?"^[42][43] Hilbert, who believed Noether's qualifications were the only important issue and that the sex of the candidate was irrelevant, objected with indignation and scolded those protesting her habilitation. Although his exact words have not been preserved, his objection is often said to have included the remark that the university was "not a bathhouse."^{[42][35][44][45]} According to Pavel Alexandrov's recollection, faculty members' opposition to Noether was based not just in sexism, but also in their objections to her social-democratic political beliefs and Jewish ancestry.^[45]

Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death is unknown. At about the same time, Noether's father retired and her brother joined the

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".[47]

Soon after arriving at Göttingen, she demonstrated her capabilities by proving the

When World War I ended, the German Revolution of 1918–1919 brought a significant change in social attitudes, including more rights for women. In 1919 the University of Göttingen allowed Noether to proceed with her habilitation (eligibility for tenure). Her oral examination was held in late May, and she successfully delivered her habilitation lecture in June 1919.^[52] Noether became a privatdozent,^[53] and she delivered that fall semester the first lectures listed under her own name.^[54] She was still not paid for her work.^[47]

Three years later, she received a letter from

Although Noether's theorem had a significant effect upon classical and quantum mechanics, among mathematicians she is best remembered for her contributions to abstract algebra. In his introduction to Noether's Collected Papers, Nathan Jacobson wrote that

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.^[1]

Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published a paper about the theory of ideals in which they defined left and right ideals in a ring.^[40]

The following year she published the paper Idealtheorie in Ringbereichen,

Van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which had become a major hub of mathematical and physical research. Russian mathematicians Pavel Alexandrov and Pavel Urysohn were the first of several in 1923.^[66] Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends.^[67] He began referring to her as der Noether, using the masculine German article as a term of endearment to show his respect. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship to Princeton University for the 1927–1928 academic year from the Rockefeller Foundation.^[67][68]

Graduate students and the Noether school

In addition to her mathematical insight, Noether was respected for her consideration of others. Although she sometimes acted rudely toward those who disagreed with her, she nevertheless gained a reputation for constant helpfulness and patient guidance of new students. Her loyalty to mathematical precision caused one colleague to name her "a severe critic", but she combined this demand for accuracy with a nurturing attitude.^[69] 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.^[61]

In Göttingen, Noether supervised more than a dozen doctoral students; her first was Grete Hermann, who defended her dissertation in February 1925. She later spoke reverently of her "dissertation-mother".^[70] Noether also supervised Max Deuring, who distinguished himself as an undergraduate and went on to contribute to the field of arithmetic geometry; Hans Fitting, remembered for Fitting's theorem and the Fitting lemma; and Zeng Jiongzhi (also rendered "Chiungtze C. Tsen" in English), who proved Tsen's theorem. She also worked closely with Wolfgang Krull, who greatly advanced commutative algebra with his Hauptidealsatz and his dimension theory for commutative rings.^[71]

Her frugal lifestyle at first was due to her being denied pay for her work. However, even after the university began paying her a small salary in 1923, she continued to live a simple and modest life. She was paid more generously later in her life, but saved half of her salary to bequeath to her nephew, Gottfried E. Noether.^[72]

Biographers suggest that she was mostly unconcerned about appearance and manners, focusing on her studies. Olga Taussky-Todd, a distinguished algebraist taught by Noether during her time at Bryn Mawr College, 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".^[73] 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.^[74]

She developed a close circle of colleagues and students interested in abstract algebra^[75] and tended to exclude those who did not. "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."^[76]

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.^[77] Later, after Nazi Germany dismissed her from teaching, she invited students into her home to discuss their plans for the future and mathematical concepts.^[78]

Influential lectures

Noether did not follow a lesson plan for her lectures, frustrating some students.^[61] She spoke quickly and her lectures were considered difficult to follow by many, including Carl Ludwig Siegel and Paul Dubreil.^[79][80] Instead, 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.^[61] Students who disliked her style often felt alienated.^[81] Despite this, she transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her.^[82][83]

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.^[63][62] Noether was recorded as having given at least five semester-long courses at Göttingen:^[84]

• 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]

These courses often preceded major publications on the same subjects.

Moscow

In the winter of 1928–1929, Noether accepted an invitation to Moscow State University, where she continued working with P.S. Alexandrov. In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry. She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory.^[85][86][87]

Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the

Ordentlicher Professor^[92][93] (full professor).^[55]

Noether's colleagues celebrated her fiftieth birthday, in 1932, in typical mathematicians' style.

noncommutative algebra are simpler than those of commutative algebra, by proving a noncommutative reciprocity law.^[94] This pleased her immensely. He also sent her a mathematical riddle, which he called the "m_μν-riddle of syllables". She solved it immediately, but the riddle has been lost.^[92][93]

In September of the same year, Noether delivered a plenary address (großer Vortrag) on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich. The congress was attended by 800 people, including Noether's colleagues Hermann Weyl, Edmund Landau, and Wolfgang 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.^[93][95]

Expulsion from Göttingen by Nazi Germany

When

Nazi activity around the country increased dramatically. At the University of Göttingen, the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by a privatdozent named Werner Weber, a former student of Noether. Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."^[96]

One of the first actions of Hitler's administration was the Law for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" by serving in World War I. In April 1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen."^[97][98] Several of Noether's colleagues, including Max Born and Richard Courant, also had their positions revoked.^[97][98]

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."^[96] Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory. When one of her students appeared in the uniform of the Nazi paramilitary organization Sturmabteilung (SA), she showed no sign of agitation and, reportedly, even laughed about it later.^[97][98]

Refuge at Bryn Mawr and Princeton

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.

Somerville College at the University of Oxford, in England. After a series of negotiations with the Rockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.^[99][100]

At Bryn Mawr, Noether met and befriended

van der Waerden's Moderne Algebra I and parts of Erich Hecke's Theorie der algebraischen Zahlen (Theory of algebraic numbers).^[103]

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of

Abraham Albert and Harry Vandiver.^[105] However, she remarked about Princeton University that she was not welcome at "the men's university, where nothing female is admitted".^[106]

Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects.[107] In the summer of 1934 she briefly returned to Germany to see Emil Artin and her brother Fritz before he left for Tomsk. Although many of her former colleagues had been forced out of the universities, she was able to use the library as a "foreign scholar".^[108][109] Without incident, Noether returned to the United States and her studies at Bryn Mawr.

Death

In April 1935 doctors discovered a

circulatory 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."^[110] She was 53.^[8]

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. Hermann Weyl and Richard Brauer traveled from Princeton and spoke with Wheeler and Taussky about their departed colleague. In the months that followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, and Pavel Alexandrov in paying their respects.^[5] Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.^[111][112]

Contributions to mathematics and physics

Noether's work in abstract algebra and topology was influential in mathematics, while Noether's theorem has widespread consequences for theoretical physics and dynamical systems. Noether showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.^[37] Her friend and colleague Hermann 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

— Weyl 1935

In the first epoch (1907–1919), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. Shortly after moving to Göttingen in 1915, she proved the two Noether's theorems, "one of the most important mathematical theorems ever proved in guiding the development of modern physics".^[10]

In the second epoch (1920–1926), Noether devoted herself to developing the theory of

noncommutative algebra, linear transformations, and commutative number fields.^[114]

Although the results of Noether's first epoch were impressive and useful, 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.

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 of ideals in rings, generalizing the earlier work of Richard Dedekind. She is also renowned for developing ascending chain conditions - a simple finiteness condition that yielded powerful results in her hands. Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as elimination theory and the algebraic varieties that had been studied by her father.

Historical context

In the century from 1832 to Noether's death in 1935, the field of mathematics – specifically

compass and straightedge. Beginning with Carl Friedrich Gauss's 1832 proof that prime numbers such as five can be factored in Gaussian integers,^[115] Évariste Galois's introduction of permutation groups in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), William Rowan Hamilton's discovery of quaternions in 1843, and Arthur 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.^[116]

Background on abstract algebra and begriffliche Mathematik (conceptual mathematics)

Two of the most basic objects in abstract algebra are groups and rings.

A group consists of a set of

associative, there must be an identity element (an element which, when combined with another element using the operation, results in the original element, such as by multiplying a number by one), and for every element there must be an inverse element.^[117][118]

A ring likewise, has a set of elements, but now has two operations. The first operation must make the set a

commutative, i.e., for any elements a and b in the ring, a + b = b + a. The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that a combined with b might be different from b combined with a. Examples of noncommutative rings include matrices and quaternions. The integers do not form a division ring, because the second operation cannot always be inverted; for example, there is no integer a such that 3a = 1.^[120][121]

The integers have additional properties which do not generalize to all commutative rings. An important example is the

Lasker–Noether theorem, for the ideals

of many rings. Much of Noether's work lay in determining what properties do hold for all rings, in devising novel analogs of the old integer theorems, and in determining the minimal set of assumptions required to yield certain properties of rings.

Groups are frequently studied through group representations. In their most general form, these consist of a choice of group, a set, and an action of the group on the set, that is, an operation which takes an element of the group and an element of the set and returns an element of the set. Most often, the set is a vector space, and the group represents symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. This is a type of symmetry of space, because space itself does not change when it is rotated even though the positions of objects in it do. Noether used these sorts of symmetries in her work on invariants in physics.

A powerful way of studying rings is through their modules. 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.^[123]

The underlying set of the module and its operation must form a group. A module is a ring-theoretic version of a group representation: Ignoring the second ring operation and the operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring itself. An important special case of this is an algebra. (The word algebra means both a subject within mathematics as well as an object studied in the subject of algebra.) An algebra consists of a choice of two rings and an operation which takes an element from each ring and returns an element of the second ring. This operation makes the second ring into a module over the first. Often the first ring is a field.

Words such as "element" and "combining operation" are very general, and can be applied to many real-world and abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For example, the elements might be

computer data words, where the first combining operation is exclusive or and the second is logical conjunction

. Theorems of abstract algebra are powerful because they are general; they govern many systems. It might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay Noether's gift to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. In his obituary of Noether, her student 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."[124]

This is the begriffliche 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.^[125]

First epoch (1908–1919)

Algebraic invariant theory

Much of Noether's work in the first epoch of her career was associated with invariant theory, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations. As an everyday example, if a rigid yardstick is rotated, the coordinates (x₁, y₁, z₁) and (x₂, y₂, z₂) of its endpoints change, but its length L given by the formula L² = Δx² + Δy² + Δz² remains the same. Invariant theory was an active area of research in the later nineteenth century, prompted in part by Felix Klein's Erlangen program, according to which different types of geometry should be characterized by their invariants under transformations, e.g., the cross-ratio of projective geometry.

An example of an invariant is the

linear operators on the vectors – typically matrices

.

The discriminant is called "invariant" because it is not changed by linear substitutions x → a x + b y, y → c x + d y with determinant a d − b c = 1 . These substitutions form the special linear group SL₂.^[c]

One can ask for all polynomials in A, B, and C that are unchanged by the action of SL₂; these are called the invariants of binary quadratic forms and turn out to be the polynomials in the discriminant.

More generally, one can ask for the invariants of homogeneous polynomials A₀ x^r y⁰ + ... + A_r x⁰ y^r of higher degree, which will be certain polynomials in the coefficients A₀, ..., A_r, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.

One of the main goals of invariant theory was to solve the "finite basis problem". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called generators, and then, adding or multiplying the generators together. For example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.

Noether's advisor, Paul 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.

special orthogonal group.^[131]

Galois theory

roots. For example, if the polynomial is x² + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of x makes the polynomial greater than or equal to one. If the field is extended

, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.

Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, +i and −i, where i is the imaginary unit, that is, i ² = −1 . More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.^[133]

The

complex conjugation, which sends +i to −i. Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a permutation of the n roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups of the Galois group.^[135]

In 1918, Noether published a paper on the

Noether's problem", which asks whether the fixed field of a subgroup G of the permutation group S_n acting on the field k(x₁, ... , x_n) always is a pure transcendental extension of the field k. (She first mentioned this problem in a 1913 paper,^[137] where she attributed the problem to her colleague Fischer.) She showed this was true for n = 2, 3, or 4. In 1969, Richard Swan found a counter-example to Noether's problem, with n = 47 and G a cyclic group of order 47^[138] (although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved.^[139]

Physics

Main articles:

Conservation law (physics), and Constant of motion

Noether was brought to

gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation 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 modern theoretical physics, with Noether's first theorem, which she proved in 1915, but did not publish until 1918.^[141] She not only solved the problem for general relativity, but also determined the conserved quantities for every system of physical laws that possesses some continuous symmetry.^[142] 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.^[143]

For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the

conservation laws of linear momentum and energy within this system, respectively.^[145]

Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool.^[7] Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon:

If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.

Second epoch (1920–1926)

Ascending and descending chain conditions

In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions.^{[citation needed]} A sequence of non-empty subsets A₁, A₂, A₃, etc. of a set S is usually said to be ascending, if each is a subset of the next

${\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \cdots .}$

Conversely, a sequence of subsets of S is called descending if each contains the next subset:

${\displaystyle A_{1}\supset A_{2}\supset A_{3}\supset \cdots .}$

A chain becomes constant after a finite number of steps if there is an n such that ${\displaystyle A_{n}=A_{m}}$ for all m ≥ n. A collection of subsets of a given set satisfies the ascending chain condition if any 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.

Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects—and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage. For example, 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. These conclusions often are crucial steps in a proof.

Many types of objects in

Noetherian space is a topological space in which every strictly ascending chain of open subspaces becomes constant after a finite number of steps; this definition makes the spectrum

of a Noetherian ring a Noetherian topological space.

The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and,

ring of formal power series

over a Noetherian ring.

Another application of such chain conditions is in

contrapositive

of the original statement. The basic premise of Noetherian induction is that every non-empty subset of S contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the minimal counterexample. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.

Commutative rings, ideals, and modules

Noether's paper, Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921),

Lasker–Noether theorem, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings. The Lasker–Noether theorem can be viewed as a generalization of the fundamental theorem of arithmetic which states that any positive integer can be expressed as a product of prime numbers

, and that this decomposition is unique.

Noether's work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields, 1927)

natural isomorphisms, and some other basic results on Noetherian and Artinian modules

.

Elimination theory

In 1923–1924, Noether applied her ideal theory to

factorization of polynomials could be carried over directly.^{[149][150][151]} Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, usually by the method of resultants

.

For illustration, a system of equations often can be written in the form

${\displaystyle \mathbf {Mv} =\mathbf {0} }$

where a matrix (or

linear transform

) M (without the variable x) times a vector ${\displaystyle \mathbf {v} }$ (that only has non-zero powers of x) is equal to the zero vector, 0. Hence, the determinant of the matrix M must be zero, providing a new equation in which the variable x has been eliminated.

Invariant theory of finite groups

Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper,

coprime to |G|! (the factorial of the order |G| of the group G). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number |G|,^[153] but Noether was not able to determine whether this bound was correct when the characteristic of the field divides |G|! but not |G|. 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.^[154][155]

In her 1926 paper,

integral

over k[x₁, ..., x_n].

Topology

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

Noether is credited with fundamental ideas that led to the development of

homology groups.^[158] According to Alexandrov, Noether attended lectures given by him and Heinz Hopf in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle"^[159]

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 the groups of algebraic complexes and cycles of a given polyhedron and the subgroup of the cycle group consisting of cycles homologous to zero; instead of the usual definition of Betti numbers, she suggested immediately defining the Betti group as the complementary (quotient) group of the group of all cycles by the subgroup of cycles homologous to zero. This observation now seems self-evident. But in those years (1925–1928) this was a completely new point of view.^[160]

Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others,

Betti group makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subject^[162] "bears the imprint of these remarks of Emmy Noether".^[163] Noether mentions her own topology ideas only as an aside in a 1926 publication,^[164] where she cites it as an application of group theory.^[165]

This algebraic approach to topology was also developed independently in

homology group, which was developed by Walther Mayer, into an axiomatic definition in 1928.^[166]

Third epoch (1927–1935)

Hypercomplex numbers and representation theory

Much work on hypercomplex numbers and group representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these results and gave the first general representation theory of groups and algebras.^[167]

Briefly, Noether subsumed the structure theory of associative algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings satisfying ascending chain conditions. This single work by Noether was of fundamental importance for the development of modern algebra.^[168]

Noncommutative algebra

Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the theory of central simple algebras.^[169]

A paper by Noether, Helmut Hasse, and

Brauer–Noether theorem^[172]

gives a characterization of the splitting fields of a central division algebra over a field.

Legacy

Noether's work continues to be relevant for the development of theoretical physics and mathematics, and she is consistently ranked as one of the greatest mathematicians of the twentieth century. In his obituary, fellow algebraist B.L. van der Waerden says that her mathematical originality was "absolute beyond comparison",^[173] and Hermann Weyl said that Noether "changed the face of [abstract] algebra by her work".^[11] Mathematician and historian Jeremy Gray wrote that any textbook on abstract algebra bears the evidence of Noether's contributions: "Mathematicians simply do ring theory her way."^[146] During her lifetime and even until today, Noether has been characterized as the greatest woman mathematician in recorded history by mathematicians^[6][174] such as Pavel Alexandrov,^[175] Hermann Weyl,^[176] and Jean Dieudonné.^[177]

In a letter to The New York Times, Albert Einstein wrote:^[5]

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.

List of doctoral students

Noether's doctoral students included:^[178]

Date	Student name	Dissertation title and English translation	University	Published
1911-12-16	Falckenberg, Hans	Verzweigungen von Lösungen nichtlinearer Differentialgleichungen Ramifications of Solutions of Nonlinear Differential Equations§	Erlangen	Leipzig 1912
1916-03-04	Seidelmann, Fritz	Die Gesamtheit der kubischen und biquadratischen Gleichungen mit Affekt bei beliebigem Rationalitätsbereich Complete Set of Cubic and Biquadratic Equations with Affect in an Arbitrary Rationality Domain§	Erlangen	Erlangen 1916
1925-02-25	Hermann, Grete	Die Frage der endlich vielen Schritte in der Theorie der Polynomideale unter Benutzung nachgelassener Sätze von Kurt Hentzelt The Question of the Finite Number of Steps in the Theory of Ideals of Polynomials using Theorems of the Late Kurt Hentzelt§	Göttingen	Berlin 1926
1926-07-14	Grell, Heinrich	Beziehungen zwischen den Idealen verschiedener Ringe Relationships between the Ideals of Various Rings§	Göttingen	Berlin 1927
1927	Doräte, Wilhelm	Über einem verallgemeinerten Gruppenbegriff On a Generalized Conceptions of Groups§	Göttingen	Berlin 1927
died before defense	Hölzer, Rudolf	Zur Theorie der primären Ringe On the Theory of Primary Rings§	Göttingen	Berlin 1927
1929-06-12	Weber, Werner	Idealtheoretische Deutung der Darstellbarkeit beliebiger natürlicher Zahlen durch quadratische Formen Ideal-theoretic Interpretation of the Representability of Arbitrary Natural Numbers by Quadratic Forms§	Göttingen	Berlin 1930
1929-06-26	Levitski, Jakob	Über vollständig reduzible Ringe und Unterringe On Completely Reducible Rings and Subrings§	Göttingen	Berlin 1931
1930-06-18	Deuring, Max	Zur arithmetischen Theorie der algebraischen Funktionen On the Arithmetic Theory of Algebraic Functions§	Göttingen	Berlin 1932
1931-07-29	Fitting, Hans	Zur Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nichtkommutativen Gruppen On the Theory of Automorphism-Rings of Abelian Groups and Their Analogs in Noncommutative Groups§	Göttingen	Berlin 1933
1933-07-27	Witt, Ernst	Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen The Riemann-Roch Theorem and Zeta Function in Hypercomplex Numbers§	Göttingen	Berlin 1934
1933-12-06	Tsen, Chiungtze	Algebren über Funktionenkörpern Algebras over Function Fields§	Göttingen	Göttingen 1934
1934	Schilling, Otto	Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper On Certain Relationships between the Arithmetic of Hypercomplex Number Systems and Algebraic Number Fields§	Marburg	Braunschweig 1935
1935	Stauffer, Ruth	The construction of a normal basis in a separable extension field	Bryn Mawr	Baltimore 1936
1935	Vorbeck, Werner	Nichtgaloissche Zerfällungskörper einfacher Systeme Non-Galois Splitting Fields of Simple Systems§	Göttingen
1936	Wichmann, Wolfgang	Anwendungen der p-adischen Theorie im Nichtkommutativen Applications of the p-adic Theory in Noncommutative Algebras§	Göttingen	Monatshefte für Mathematik und Physik (1936) 44, 203–24.

See also

• List of things named after Emmy Noether
• Timeline of women in science

Notes

1. ^
   University of Erlangen in 1907.^[1][2] Sometimes Emmy is mistakenly reported as a short form for Amalie, or misreported as Emily. Lee Smolin wrote "Emily Noether, a great German mathematician" in a letter for The Reality Club.^[3]
2. ^ Lederman & Hill 2004, p. 71 write that she completed her doctorate at Göttingen, but this appears to be an error.
3. ^ There are no invariants under the general linear group of all invertible linear transformations because these transformations can be multiplication by a scaling factor. To remedy this, classical invariant theory also considered relative invariants, which were forms invariant up to a scale factor.

References

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4. ^ Conover, Emily (12 June 2018). "In her short life, mathematician Emmy Noether changed the face of physics". Science News. Retrieved 2 July 2018.
5. ^ ^{a b c} Einstein, Albert (1 May 1935), "Professor Einstein Writes in Appreciation of a Fellow-Mathematician", The New York Times (published 4 May 1935), retrieved 13 April 2008. Transcribed online at the MacTutor History of Mathematics Archive.
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8. ^ ^{a b c d} Ogilvie & Harvey 2000
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11. ^ ^{a b} Dick 1981, p. 128
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13. ^ Dick 1981, pp. 9–10.
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17. ^ Dick 1981, pp. 25, 45.
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31. ^ Dick 1981, pp. 18, 24.
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33. ^ Dick 1981, p. 23.
34. ^ Rowe 2021, p. 22.
35. ^ ^{a b c d} Weyl 1935.
36. ^ Dick 1981, pp. 23–24.
37. ^ ^{a b} Kimberling 1981, pp. 11–12.
38. ^ Dick 1981, pp. 18–24.
39. ^ Rowe 2021, pp. 29–35.
40. ^ ^{a b} Rowe & Koreuber 2020, p. 27.
41. ^ O'Connor, John J.; Robertson, Edmund F., "Emmy Noether's doctoral students", MacTutor History of Mathematics Archive, University of St Andrews
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43. ^ ^{a b} Lederman & Hill 2004, p. 72.
44. ^ Rowe & Koreuber 2020, pp. 75–76.
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46. ^ Dick 1981, pp. 24–26.
47. ^ ^{a b} Byers 2006, pp. 91–92.
48. ^ Byers 2006, p. 86.
49. ^ Noether 1918c, p. 235.
50. ^ Rowe & Koreuber 2020, p. 3.
51. ^ Byers 1996, p. 2.
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53. ^ Kosmann-Schwarzbach 2011, p. 49.
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55. ^ ^{a b} Dick 1981, p. 188.
56. ^ Kimberling 1981, pp. 14–18.
57. ^ Dick 1981, pp. 33–34.
58. ^ ^{a b} Noether 1921.
59. ^ ^{a b} Kimberling 1981, p. 18.
60. ^ Dick 1981, pp. 44–45.
61. ^ ^{a b c d} van der Waerden 1935.
62. ^ ^{a b c d} O'Connor, John J.; Robertson, Edmund F., "Emmy Amalie Noether", MacTutor History of Mathematics Archive, University of St Andrews
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66. ^ Kimberling 1981, p. 24.
67. ^ ^{a b} Kimberling 1981, pp. 24–25.
68. ^ Dick 1981, pp. 61–63.
69. ^ Dick 1981, pp. 37–49.
70. ^ Dick 1981, p. 51.
71. ^ Dick 1981, pp. 53–57.
72. ^ Dick 1981, pp. 46–48.
73. ^ Taussky 1981, p. 80.
74. ^ Dick 1981, pp. 40–41.
75. ^ Rowe & Koreuber 2020, p. 32.
76. ^ Dick 1981, pp. 38–41.
77. ^ Mac Lane 1981, p. 71.
78. ^ Dick 1981, p. 76.
79. ^ Rowe & Koreuber 2020, p. 21, 122.
80. ^ Dick 1981, pp. 37–38.
81. ^ Mac Lane 1981, p. 77.
82. ^ Rowe & Koreuber 2020, p. 36, 99.
83. ^ Dick 1981, p. 38.
84. ^ Scharlau, Winfried, Emmy Noether's Contributions to the Theory of Algebras in Teicher 1999, p. 49.
85. ^ Dick 1981, pp. 63–64.
86. ^ Kimberling 1981, p. 26.
87. ^ Alexandrov 1981, pp. 108–10.
88. ^ ^{a b} Alexandrov 1981, pp. 106–09.
89. ^ McLarty 2005.
90. ^ Dick 1981, pp. 82–83.
91. ^ O'Connor, John J.; Robertson, Edmund F., "Fritz Alexander Ernst Noether", MacTutor History of Mathematics Archive, University of St Andrews.
92. ^ ^{a b} Dick 1981, pp. 72–73.
93. ^ ^{a b c} Kimberling 1981, pp. 26–27.
94. ^ Hasse 1933, p. 731.
95. ^ Dick 1981, pp. 74–75.
96. ^ ^{a b} Kimberling 1981, p. 29.
97. ^ ^{a b c} Dick 1981, pp. 75–76.
98. ^ ^{a b c} Kimberling 1981, pp. 28–29.
99. ^ Dick 1981, pp. 78–79.
100. ^ Kimberling 1981, pp. 30–31.
101. ^ Kimberling 1981, pp. 32–33.
102. ^ Dick 1981, p. 80.
103. ^ Dick 1981, pp. 80–81.
104. ^ "Emmy Noether at the Institute for Advanced Study". StoryMaps. ArcGIS. 7 December 2019. Retrieved 28 August 2020.
105. ^ Dick 1981, pp. 81–82.
106. ^ Dick 1981, p. 81.
107. ^ Dick 1981, p. 83.
108. ^ Dick 1981, p. 82.
109. ^ Kimberling 1981, p. 34.
110. ^ ^{a b} Kimberling 1981, pp. 37–38.
111. ^ Kimberling 1981, p. 39.
112. ^ Alan Chodos, ed. (March 2013). "This Month in Physics History: March 23, 1882: Birth of Emmy Noether". APS News. American Physical Society. Retrieved 28 August 2020. (Volume 22, Number 3)
113. ^ Gilmer 1981, p. 131.
114. ^ Kimberling 1981, pp. 10–23.
115. ^ Gauss, C.F. (1832). "Theoria residuorum biquadraticorum – Commentatio secunda". Comm. Soc. Reg. Sci. Göttingen (in Latin). 7: 1–34. reprinted in Werke [Complete Works of C.F. Gauss]. Hildesheim: Georg Olms Verlag. 1973. pp. 93–148.
116. ^ G.E. Noether 1987, p. 168.
117. ^ Lang 2005, p. 16, II.§1.
118. ^ Stewart 2015, pp. 18–19.
119. ^ Stewart 2015, p. 182.
120. ^ Stewart 2015, p. 183.
121. ^ Gowers et al. 2008, p. 284.
122. ^ Gowers et al. 2008, pp. 699–700.
123. ^ Gowers et al. 2008, p. 285.
124. ^ Dick 1981, p. 101.
125. ^ Gowers et al. 2008, p. 801.
126. ^ Noether 1908.
127. ^ Noether 1914, p. 11.
128. ^ Gordan 1870.
129. ^ Weyl 1944, pp. 618–21.
130. ^ Hilbert 1890, p. 531.
131. ^ Hilbert 1890, p. 532.
132. ^ Stewart 2015, p. 108–111
133. ^ Stewart 2015, p. 129–130
134. ^ Stewart 2015, p. 112–114
135. ^ Stewart 2015, p. 114–116; 151–153
136. ^ Noether 1918.
137. ^ Noether 1913.
138. ^ Swan 1969, p. 148.
139. ^ Malle & Matzat 1999.
140. ^ Gowers et al. 2008, p. 800.
141. ^ Noether 1918b
142. ^ Lynch, Peter (18 June 2015). "Emmy Noether's beautiful theorem". ThatsMaths. Retrieved 28 August 2020.
143. ^ Kimberling 1981, p. 13
144. ^ Lederman & Hill 2004, pp. 97–116.
145. ^ Angier, Natalie (26 March 2012). "The Mighty Mathematician You've Never Heard Of". The New York Times. Retrieved 28 August 2020.
146. ^ ^{a b} Gray 2018, p. 294.
147. ^ ^{a b} Gilmer 1981, p. 133.
148. ^ Noether 1927.
149. ^ Noether 1923.
150. ^ Noether 1923b.
151. ^ Noether 1924.
152. ^ Noether 1915.
153. ^ Fleischmann 2000, p. 24.
154. ^ Fleischmann 2000, p. 25.
155. ^ Fogarty 2001, p. 5.
156. ^ Noether 1926.
157. ^ Haboush 1975.
158. ^ Hilton 1988, p. 284.
159. ^ Dick 1981, p. 173.
160. ^ ^{a b} Dick 1981, p. 174.
161. ^ Hirzebruch, Friedrich, Emmy Noether and Topology in Teicher 1999, p. 57–61
162. ^ Hopf 1928.
163. ^ Dick 1981, pp. 174–75.
164. ^ Noether 1926b.
165. ^ Hirzebruch, Friedrich, Emmy Noether and Topology in Teicher 1999, p. 63
166. ^ Hirzebruch, Friedrich, Emmy Noether and Topology in Teicher 1999, p. 61-63
167. ^ Noether 1929.
168. ^ van der Waerden 1985, p. 244.
169. ^ Lam 1981, pp. 152–53.
170. ^ Brauer, Hasse & Noether 1932.
171. ^ Noether 1933.
172. ^ Brauer & Noether 1927.
173. ^ Dick 1981, p. 100.
174. ^ James 2002, p. 321.
175. ^ Dick 1981, p. 154.
176. ^ Dick 1981, p. 152.
177. ^ Noether 1987, p. 167.
178. ^ Emmy Noether at the Mathematics Genealogy Project

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Selected works by Emmy Noether

Main article: Emmy Noether bibliography

• Noether, Emmy (1908), "Über die Bildung des Formensystems der ternären biquadratischen Form" [On Complete Systems of Invariants for Ternary Biquadratic Forms], Journal für die Reine und Angewandte Mathematik (in German), 1908 (134),
  S2CID 119967160, archived from the original
  on 8 March 2013
• ——— (1913), "Rationale Funktionenkörper" [Rational Function Fields], J. Ber. D. DMV (in German), 22, DE: 316–19, archived from the original on 8 March 2013
• ——— (1915), "Der Endlichkeitssatz der Invarianten endlicher Gruppen" [The Finiteness Theorem for Invariants of Finite Groups] (PDF),
  S2CID 121213008
• ——— (1918), "Gleichungen mit vorgeschriebener Gruppe" [Equations with Prescribed Group],
  S2CID 122353858, archived from the original
  on 3 September 2014
• ——— (1918b). "Invariante Variationsprobleme" [Invariant Variation Problems]. Nachr. D. König. Gesellsch. D. Wiss. (in German). 918 (3). Translated by Tavel, M.A. Göttingen: 235–57.
  S2CID 119019843
  .
• ——— (1918c). "Invariante Variationsprobleme" [Invariant Variation Problems]. Nachr. D. König. Gesellsch. D. Wiss. (in German). 918. Göttingen: 235–57. Archived from the original on 5 July 2008. Original German image with link to Tavel's English translation
• ——— (1921), "Idealtheorie in Ringbereichen" [The Theory of Ideals in Ring Domains],
  S2CID 121594471, archived from the original
  (PDF) on 3 September 2014

• Berlyne, Daniel (11 January 2014). "Ideal Theory in Rings (Translation of "Idealtheorie in Ringbereichen" by Emmy Noether)".
  arXiv:1401.2577 [math.RA
  ].

• ——— (1923), "Zur Theorie der Polynomideale und Resultanten" (PDF),
  S2CID 122226025
• ——— (1923b), "Eliminationstheorie und allgemeine Idealtheorie" (PDF),
  S2CID 121239880
• ——— (1924), "Eliminationstheorie und Idealtheorie", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 33, DE: 116–20, archived from the original on 8 March 2013
• ——— (1926), "Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p" [Proof of the Finiteness of the Invariants of Finite Linear Groups of Characteristic p], Nachr. Ges. Wiss (in German), DE: 28–35, archived from the original on 8 March 2013
• ——— (1926b), "Ableitung der Elementarteilertheorie aus der Gruppentheorie" [Derivation of the Theory of Elementary Divisor from Group Theory], Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 34 (Abt. 2), DE: 104, archived from the original on 8 March 2013
• ——— (1927), "Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern" [Abstract Structure of the Theory of Ideals in Algebraic Number Fields],
  S2CID 121288299, archived from the original
  (PDF) on 3 September 2014
• Brauer, Richard; Noether, Emmy (1927), "Über minimale Zerfällungskörper irreduzibler Darstellungen" [On the Minimum Splitting Fields of Irreducible Representations], Sitz. Ber. D. Preuss. Akad. D. Wiss. (in German): 221–28
• Noether, Emmy (1929), "Hyperkomplexe Größen und Darstellungstheorie" [Hypercomplex Quantities and the Theory of Representations],
  S2CID 120464373, archived from the original
  on 29 March 2016
• Brauer, Richard;
  S2CID 199545542
• Noether, Emmy (1933), "Nichtkommutative Algebren" [Noncommutative Algebras], Mathematische Zeitschrift (in German), 37: 514–41,
  S2CID 186227754
• ——— (1983),
  MR 0703862

Further reading

Library resources about
Emmy Noether

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• Resources in your library
• Resources in other libraries

By Emmy Noether

• Online books
• Resources in your library
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Articles

• Angier, Natalie (26 March 2012). "The Mighty Mathematician You've Never Heard Of". The New York Times. Retrieved 27 January 2024.
• Phillips, 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:
  ISSN 0002-9920
  .

Online biographies

• Byers, Nina (16 March 2001), "Emmy Noether", Contributions of 20th Century Women to Physics, UCLA, archived from the original on 12 February 2008, retrieved 27 January 2024.
• Taylor, Mandie (22 February 2023), "Emmy Noether", Biographies of Women Mathematicians, Agnes Scott College, retrieved 27 January 2024.

External links

• Emmy Noether at the Mathematics Genealogy Project
• Noether's application for admission to the University of Erlangen and three of her curriculum vitae from the website of historian Cordula Tollmien
• Letter by Noether to Dr. Park, Bryn Mawr College President – Bryn Mawr College Library Special Collections
• Photograph of Noether taken by Hanna Kunsch – Bryn Mawr College Library Special Collections
• Photographs of Noether – Oberwolfach Photo Collection of the
  Mathematisches Forschungsinstitut Oberwolfach
• Photographs of Noether's colleagues and acquaintances from the website of Clark Kimberling
• Emmy Noether on In Our Time at the BBC