Emmy Noether

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 to a Jewish family in the Franconian town of Erlangen; her father was the mathematician Max Noether. She originally planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen–Nuremberg, where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research. The philosophical faculty objected, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain the rank of Privatdozent.

Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called the "Noether Boys". In 1924, Dutch mathematician 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 graduate and post-doctoral women including Marie Johanna Weiss and Olga Taussky-Todd. At the same time, she lectured and conducted research at the Institute for Advanced Study in Princeton, New Jersey.

Noether's mathematical work has been divided into three "epochs". In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields. Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra". 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.

Education

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

This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing mixed-sex education would "overthrow all academic order". One of just two women in a university of 986 students, Noether was allowed only to audit classes rather than participate fully, and she required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903, she passed the graduation exam at a Realgymnasium in Nuremberg.

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.

thumb|[[Paul Gordan supervised Noether's doctoral dissertation on invariants of biquadratic forms.]]

In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded. 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. 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. 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. It had been well received, but Noether later described her thesis and some subsequent similar papers she produced as "crap". All of her later work was in a completely different field.

University of Erlangen–Nuremberg

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. She joined the Circolo Matematico di Palermo in 1908 and the Deutsche Mathematiker-Vereinigung in 1909. In 1910 and 1911, she published an extension of her thesis work from three variables to n variables.

thumb|upright=1.2|Noether sometimes used postcards to discuss abstract algebra with her colleague, [[Ernst Sigismund Fischer|Ernst Fischer. This card is postmarked 10 April 1915.]]

Gordan retired in 1910, and Noether taught under his successors, Erhard Schmidt and Ernst Fischer, who took over from the former in 1911. 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. 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.

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. This phase marked Noether's first exposure to abstract algebra, the field to which she would make groundbreaking contributions.

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. Schilling also began studying under Noether, but was forced to find a new advisor due to Noether's emigration. Under Helmut Hasse, he completed his PhD in 1934 at the University of Marburg. He later worked as a post doc at Trinity College, Cambridge, before moving to the United States. Noether was recorded as having given at least 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

thumb|Noether taught at [[Moscow State University in 1928–1929.]]

thumb|upright=0.6|[[Pavel Alexandrov]]

In 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.

Politics was not central to her life, but Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution. She was especially happy to see Soviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from a pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess". Hermann Weyl recalled that "During the wild times after the Revolution of 1918," Noether "sided more or less with the Social Democrats". She was a member of the Independent Social Democrats from 1919 to 1922, a short-lived splinter party. In the words of logician and historian Colin McLarty, "she was not a Bolshevist, but was not afraid to be called one."

Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933, he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry. This proved unsuccessful, but they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union.

Recognition

In 1932, Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics. She also worked with Abraham Albert and Harry Vandiver. She remarked about Princeton University that she was not welcome at "the men's university, where nothing female is admitted".

Her time in the United States was pleasant, as she was surrounded by supportive colleagues and absorbed in her favorite subjects. In mid-1934, she briefly returned to Germany to see Emil Artin and her brother Fritz. The latter, after having been forced out of his job at the Technische Hochschule Breslau, had accepted a position at the Research Institute for Mathematics and Mechanics in Tomsk, in the Siberian Federal District of Russia.

Many of her former colleagues had been forced out of the universities, but she was able to use the library in Göttingen as a "foreign scholar". Without incident, Noether returned to the United States and her studies at Bryn Mawr.

Death

In April 1935, doctors discovered a tumor in Noether's pelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst "the size of a large cantaloupe". Two smaller tumors in her uterus appeared to be benign and were not removed to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse on the fourth. On 14 April, Noether fell unconscious, her temperature soared to , and she died. "[I]t is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located." She was 53.

thumb|right|Noether's ashes were placed under the cloistered walkway of Bryn Mawr's [[Bryn Mawr College#Old Library (previously M. Carey Thomas Library and College Hall)|Old Library.]]

A few days after Noether's death, her friends and associates at Bryn Mawr held a small memorial service at College President Park's house. Hermann Weyl and Richard Brauer both traveled from Princeton and delivered eulogies. 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.

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. Her friend and colleague Hermann Weyl described her scholarly output in three epochs:

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".

In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings. In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields. The results of Noether's first epoch were impressive and useful, but her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B. L. van der Waerden in their obituaries of her.

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 conditionsa 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 the topics of algebraic invariants that had been studied by her father and elimination theory, discussed below.

Noether's most important contributions to mathematics were to the development of an emerging new field, abstract algebra.

Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. In his obituary of Noether, van der Waerden recalled that

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.

First epoch (1908–1919)

Algebraic invariant theory

thumb|right|Table 2 from Noether's dissertation on invariant theory. This table collects 202 of the 331 invariants of ternary biquadratic forms. These forms are graded in two variables x and u. The horizontal direction of the table lists the invariants with increasing grades in x, while the vertical direction lists them with increasing grades in u.

Much of Noether's work in the first epoch of her career was associated with invariant theory, principally algebraic invariant theory. Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations. As an everyday example, if a rigid metre-stick is rotated, the coordinates of its endpoints change, but its length remains the same. A more sophisticated example of an invariant is the discriminant of a homogeneous quadratic polynomial , where and are indeterminates. The discriminant is called "invariant" because it is not changed by linear substitutions and with determinant . These substitutions form the special linear group .

One can ask for all polynomials in , , and that are unchanged by the action of ; these turn out to be the polynomials in the discriminant. More generally, one can ask for the invariants of homogeneous polynomials of higher degree, which will be certain polynomials in the coefficients , 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 a quadratic polynomial.

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. He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables. Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.

Noether followed Gordan's lead, writing her doctoral dissertation and several other publications on invariant theory. She extended Gordan's results and also built upon Hilbert's research. Later, she would disparage this work, finding it of little interest and admitting to forgetting the details of it. Hermann Weyl wrote, <blockquote>[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.</blockquote>

Galois theory

Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable of degree , in which the coefficients are drawn from some ground field, which might be, for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not be choices of , which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. For example, if the polynomial is and the field is the real numbers, then the polynomial has no roots, because any choice of makes the polynomial greater than or equal to one. If the field is extended 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, and , where is the imaginary unit, that is, More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.

The Galois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial. (These transformations are called automorphisms.) The Galois group of consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends to . Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, so transformation determines a permutation of the 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.

In 1918, Noether published a paper on the inverse Galois problem. Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "Noether's problem", which asks whether the fixed field of a subgroup G of the permutation group acting on the field always is a pure transcendental extension of the field . (She first mentioned this problem in a 1913 paper, where she attributed the problem to her colleague Fischer.) She showed this was true for = 2, 3, or 4. In 1969, Richard Swan found a counter-example to Noether's problem, with = 47 and a cyclic group of order 47 (although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved.

Physics

Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, because gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, in a 1918 paper. This paper presented two theorems, of which the first is known as Noether's theorem. Together, these theorems not only solve the problem for general relativity, but also determine the conserved quantities for every system of physical laws that possesses some continuous symmetry. Upon receiving her work, Einstein wrote to Hilbert:

For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the system is responsible for the conservation law. As another example, if a physical experiment works the same way at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.

At the time, physicists were not familiar with Sophus Lie's theory of continuous groups, on which Noether had built. Many physicists first learned of Noether's theorem from an article by Edward Lee Hill that presented only a special case of it. Consequently, the full scope of her result was not immediately appreciated. During the latter half of the 20th century, Noether's theorem became a fundamental tool of modern theoretical physics, because of the insight it gives into conservation laws, and also as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon: If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments.

Commutative rings, ideals, and modules

Noether's paper, Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921), is the foundation of general commutative ring theory, and gives one of the first general definitions of a commutative ring. Before her paper, most results in commutative algebra were restricted to special examples of commutative rings, such as polynomial rings over fields or rings of algebraic integers. Noether proved that in a ring which satisfies the ascending chain condition on ideals, every ideal is finitely generated. In 1943, French mathematician Claude Chevalley coined the term Noetherian ring to describe this property. A major result in Noether's 1921 paper is the Lasker–Noether theorem, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings. The Lasker–Noether theorem can be viewed as a generalization of the fundamental theorem of arithmetic which states that any positive integer can be expressed as a product of prime 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) characterized the rings in which the ideals have unique factorization into prime ideals (now called Dedekind domains). Noether showed that these rings were characterized by five conditions: they must satisfy the ascending and descending chain conditions, they must possess a unit element, but no zero divisors, and they must be integrally closed in their associated field of fractions. This paper also contains what now are called the isomorphism theorems, which describe some fundamental natural isomorphisms, and some other basic results on Noetherian and Artinian modules.

Elimination theory

In 1923–1924, Noether applied her ideal theory to elimination theory in a formulation that she attributed to her student, Kurt Hentzelt. She showed that fundamental theorems about the factorization of polynomials could be carried over directly.

Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, often by the method of resultants.

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

where a matrix (or linear transform) (without the variable ) times a vector (that only has non-zero powers of ) is equal to the zero vector, . Hence, the determinant of the matrix must be zero, providing a new equation in which the variable 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, Noether found a solution to the finite basis problem for a finite group of transformations acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is called Noether's bound. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field is coprime to <math>\left|G\right|!</math> (the factorial of the order <math>\left|G\right|</math> of the group ). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number <math>\left|G\right|</math>, but Noether was not able to determine whether this bound was correct when the characteristic of the field divides <math>\left|G\right|!</math> but not <math>\left|G\right|</math>. 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.

In her 1926 paper, Noether extended Hilbert's theorem to representations of a finite group over any field; the new case that did not follow from Hilbert's work is when the characteristic of the field divides the order of the group. Noether's result was later extended by William Haboush to all reductive groups by his proof of the Mumford conjecture. In this paper Noether also introduced the Noether normalization lemma, showing that a finitely generated domain over a field has a set } of algebraically independent elements such that is integral over .

Topology

thumb|right|A continuous deformation ([[homotopy) of a coffee cup into a doughnut (torus) and back]]

As noted by Hermann Weyl in his obituary, 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 algebraic topology from the earlier combinatorial topology, specifically, the idea of homology groups. According to Alexandrov, Noether attended lectures given by him and Heinz Hopf in 1926 and 1927, where "she continually made observations which were often deep and subtle" and he continues that,

Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others, and it became a frequent topic of discussion among the mathematicians of Göttingen. Noether observed that her idea of a Betti group makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subject "bears the imprint of these remarks of Emmy Noether". Noether mentions her own topology ideas only as an aside in a 1926 publication, where she cites it as an application of group theory.

This algebraic approach to topology was also developed independently in Austria. In a 1926–1927 course given in Vienna, Leopold Vietoris defined a homology group, which was developed by Walther Mayer into an axiomatic definition in 1928.

thumb|upright|right|[[Helmut Hasse worked with Noether and others to found the theory of central simple algebras.]]

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 earlier results and gave the first general representation theory of groups and algebras. This single work by Noether was said to have ushered in a new period in modern algebra and to have been of fundamental importance for its development.

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.

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.

A paper by Noether, Hasse, and Brauer pertains to division algebras, which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a finite-dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is trivial), and from this, deduced their Hauptsatz ("main theorem"):<blockquote>Every finite-dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension.</blockquote>These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra are splitting fields. This paper also contains the Skolem–Noether theorem, which states that any two embeddings of an extension of a field into a finite-dimensional central simple algebra over are conjugate. The Brauer–Noether theorem gives a characterization of the splitting fields of a central division algebra over a field.

Legacy

thumb|right|The Emmy–Noether–Campus at the [[University of Siegen is home to its mathematics and physics departments.]]

thumb|upright|right|Noether is one of the carved stone busts displayed in Germany's [[Ruhmeshalle München (or Munich Hall of Fame in English).]]

Noether's work continues to be relevant for the development of theoretical physics and mathematics, and she is considered one of the most important mathematicians of the twentieth century. During her lifetime and even until today, Noether has also been characterized as the greatest woman mathematician in recorded history by mathematicians such as Pavel Alexandrov, Hermann Weyl, and Jean Dieudonné.

In a letter to The New York Times, Albert Einstein wrote: 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." Several things now bear her name, including many mathematical objects, and an asteroid, 7001 Noether. In 2019 Time created 89 new covers to celebrate women of the year starting from 1920; it chose Noether for 1921.

Notes

References

Sources

. Reprinted in .
. Reprinted as an appendix in .

Selected works by Emmy Noether

Original German image with link to Tavel's English translation

External links

Papers

Noether's application for admission to the University of Erlangen–Nuremberg and three of her curriculum vitae from the Web site of historian
Letter by Noether to Marion Edwards Park, Bryn Mawr College President — Bryn Mawr College Library Special Collections

Media

(audio)
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 Web site of Clark Kimberling