Emmy Noether

Amalie Emmy Noether, (March 23, 1882 – April 14, 1935) was a German-born Jewish mathematician who is known mainly for her seminal contributions to abstract algebra. Often described as the most important woman mathematician of all time,[1][2] she revolutionized the mathematical theories of rings and fields, as well as commutative and noncommutative algebras. In modern theoretical physics, her two Noether's theorems, which explain the connections between symmetry and conservation laws, are essential tools of research.[3]

Born in the Bavarian town of Erlangen to the noted mathematician Max Noether and his wife, Emmy showed intellectual promise at a young age. Although she passed the examinations required to teach French and English, she continued her studies in mathematics at the University of Erlangen, where her father lectured. After completing her dissertation in 1907 under the supervision of Paul Albert Gordan, she worked at the Mathematical Institute without pay for seven years.

In 1915 she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen. The Philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her Habilitation process was approved after that time, paving the way for her to obtain the rank of Privatdozent. She spent the next fourteen years gaining respect for her groundbreaking mathematics work, culminating with a major address at the 1932 International Congress of Mathematicians in Zürich, Switzerland. The following year, Germany's Nazi government fired her from Göttingen, and she moved to the United States, where she took a position at Bryn Mawr College in Pennsylvania. In 1935 she underwent surgery for an ovarian cyst and, despite signs of speedy recovery, died four days later at the age of 53.

Most of Noether's mathematical work was focused on algebra, and has been divided into three "epoch"s. In the first epoch (1908–1919), she made important contributions to invariant theory, most notably her famous Noether's theorem, which has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics".[4] In the second epoch (1920–1926), Noether studied commutative algebras, most notably working out the theory of ideals in rings. Her paper Idealtheorie in Ringbereichen (1921) is considered a classic of mathematics, introducing Noetherian rings and proving several isomorphism theorems about them. Noether is known for her elegant use of ascending chain conditions, which is recognized in Noetherian modules. In the third epoch (1927–1935), she worked on noncommutative algebras and on uniting hypercomplex numbers and the representation theory of groups with the modules and ideals of rings. She is credited with formulating a theory of modules and ideals in rings that satisfy certain finiteness conditions. Her papers Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (1927), Hyperkomplexe Größen und Darstellungstheorie (1929) and Beweis eines Hauptsatzes in der Theorie der Algebren (1932) are considered milestones of modern algebraic theory. Noether was also generous with her ideas, and is credited with several novel lines of research by other mathematicians, even in disparate fields such as algebraic topology.

Biography

The Noether family had descended from Jewish wholesale traders in Germany. Emmy's father, Max Noether, was an important mathematician who was largely self-taught. He had been paralyzed by polio at the age of fourteen and even after regaining mobility was handicapped in one leg. He received a doctorate from the University of Heidelberg in 1868. After teaching there for seven years, he took a position in the Bavarian city of Erlangen, where he met and married Ida Amalia Kaufmann, the daughter of a prosperous Jewish merchant.[5] As a mathematician, Max Noether contributed mainly to algebraic geometry, following in the footsteps of Alfred Clebsch. He is known primarily for his "residue" or "AF+BG" theorem, but he produced several others, such as the Brill–Noether theorem.

Emmy Noether was born on 23 March 1882, the first of four children. Her first name was Amalie, after her mother and paternal grandmother, but she began using her middle name at a young age. As a girl, she was well-liked, although she did not stand out academically. Near-sighted and talking during childhood with a minor lisp, she was known for being clever and friendly. A family friend recounted a story years later about young Emmy quickly solving a brain teaser at a children's party, showing logical acumen at an early age.[6] Emmy was taught to cook and clean – like most girls of the time – and took lessons on the piano. She pursued none of these activities with passion, although she loved to dance.[7]

Of her three brothers, only Fritz Noether, born in 1884, is remembered for his academic accomplishments. Alfred, born in 1883, received a doctorate in chemistry from Erlangen in 1909; he died nine years later. The family's youngest child, Gustav Robert, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928. Fritz, however, made a reputation for himself in the field of applied mathematics after studying in Munich.[8]

University of Erlangen

Emmy Noether showed an 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 at girls' schools, but she chose instead to continue her studies at the University of Erlangen. This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing coeducation would "overthrow all academic order".[9] One of only two females in a school of 986, Noether was forced to audit classes and required to gain the permission of individual professors whose lectures she wished to attend. Despite the obstacles, on 14 July 1903 she passed the graduation exam at a Realgymnasium in Nuremberg.[10]

During the winter semester of 1903–04, she studied as an auditor at the University of Göttingen, attending lectures from astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert. Soon thereafter, the law restricting women's rights in the university was rescinded, and Noether returned to Erlangen. She officially entered the school on 24 October 1904, and declared her intention to focus solely on mathematics. Working under the tutelage of Paul Albert Gordan, in 1907 she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form ("On Complete Systems of Invariants for Ternary Biquadratic Forms"). Although it was well received, Noether later referred to her thesis as "crap" and "a jungle of formulas".[11]

For the next seven years she taught at the University of Erlangen's Mathematical Institute, without pay. Continuing her research on invariant theory, she occasionally substituted for her father when he was too ill to lecture. She also worked with Erhard Schmidt and Ernst Fischer, sometimes discussing advanced concepts with the latter through the mail by writing commentary on postcards.[12]

University of Göttingen

In 1915, David Hilbert invited Emmy Noether to join the mathematics department at the University of Göttingen, challenging the objections of his colleagues that a woman should not be allowed to teach mathematics to university students.

In 1915, David Hilbert invited Emmy Noether to join the mathematics department at the University of Göttingen, challenging the objections of his colleagues that a woman should not be allowed to teach mathematics to university students.

In the spring of 1915 Noether was invited by David Hilbert and Felix Klein to return to the University of Göttingen. Their effort to recruit her was blocked, however, by the philologists and historians in the Philosophical faculty; women, they insisted, should not be hired in the role of Privatdozent. One oppositional colleague 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?"[13] Hilbert responded to this question with indignation: "I do not see that the sex of the candidate is an argument against her admission as Privatdozent," he said. "After all, we are a university, not a bath house."[13]

Two weeks after Noether left for Göttingen, her mother died suddenly in Erlangen. She had previously received medical care for an eye malady, but its nature and impact on her death is unknown. Around the same time, Noether's father retired and her brother joined the army to serve in World War I. She returned to Erlangen for several weeks, mostly to care for her aging father.[14]

During her first years at Göttingen, she worked in an unpaid and undefined role; her family paid for her room and board, and supported her academic work. Her lectures were often advertised under Hilbert's name, with Noether providing "assistance". Soon after arriving, however, she demonstrated her value to the department by proving Noether's theorem, which shows that a conservation law can be derived from any differentiable symmetry of a physical system.[15] US physicists Leon Lederman and Christopher T. Hill, in their book Symmetry and the Beautiful Universe, argue that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem."[4]

When World War I ended, the German Revolution of 1918-19 brought a significant change in social attitudes, including more rights for women. In 1919 the University of Göttingen allowed her to proceed with the Habilitation, a process to obtain the rank of Privatdozent. Her oral examination took place in late May, and she successfully delivered her Habilitation lecture in June. Three years later she received a letter from the Prussian Minister for Science, Art, and Public Education, in which he presented her the title of nicht beamteter ausserordentlicher Professor ("unofficial associate professor"). Although it recognized the importance of her work, the position still did not provide her with a salary. Not until she was appointed to the special position of Lehrauftrag für Algebra a year later did she receive for her lectures.[16]

In 1920 Noether collaborated with a colleague named W. Schmeidler on a paper about the theory of ideals. Their work was the first to define left and right ideals. One year later she published a landmark paper called Idealtheorie in Ringbereichen, which analyzed ascending chain conditions with regard to ideals. Canadian mathematician Irving Kaplansky called this work "revolutionary",[17] and its importance is seen in the use of the term "Noetherian ring" to describe a ring which adheres to ascending chain conditions.[18]

Soon afterwards, she began supervising doctoral students, including Grete Hermann, who later spoke reverently of her "dissertation-mother".[19] Noether also sponsored the doctoral work of Max Deuring, who distinguished himself as an undergraduate and went on to contribute significantly to the field of arithmetic geometry; Hans Fitting, who established Fitting's theorem as well as the Fitting lemma; and Zeng Jiongzhi, who proved Tsen's theorem. She also worked closely with Wolfgang Krull, originator of Krull's theorem.[20]

Dutch mathematician Bartel Leendert van der Waerden came to the University of Göttingen in 1924. He began working immediately with Noether, who provided invaluable methods of abstract conceptualization. He said later that her originality was "absolute beyond comparison".[21] In 1931 he published Moderne Algebra, a central text in the field; its second volume borrows heavily from Noether's work. He acknowledged his debt to her in a note for the seventh edition reading "Based in part on lectures by E. Artin and E. Noether".[22] In other instances she allowed her colleagues and students to receive credit for her ideas, helping them develop their careers rather than demanding tribute.[23]

Van der Waerden's visit was part of an international convergence on Göttingen, which became a central hub of activity among mathematicians worldwide. From 1926 to 1930, Russian topologist Pavel Alexandrov lectured at the university, and quickly became good friends with Noether. He began referring to her as der Noether, using the masculine article as a term of endearment to show his respect. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was only able to help him secure a scholarship from the Rockefeller Foundation.[24] They met regularly and enjoyed ongoing discussions about the intersections of algebra and topology. In his 1935 memorial address, Alexandrov named her "the greatest woman mathematician of all time".[25]

Moscow

In the winter of 1928–29, Noether accepted an invitation to the University of Moscow, where she continued working with Alexandrov and his colleagues. In addition to her own research work, she taught classes in abstract algebra and algebraic geometry. She worked with the topologist Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory.[26]

Although politics were not central to her life, 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 advancements in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from a pension building, after student leaders complained of living with "a Marxist-leaning Jewess".[27]

After her time at the University of Moscow, Noether planned to return – an effort for which she received support from Alexandrov. After leaving Germany in 1933, he tried to help her gain a chair at Moscow through the Commissariat of Education. Although this effort was unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union.[27] Her brother Fritz, meanwhile, took a position at the Research Institute for Mathematics and Mechanics in Tomsk, Siberia after losing his job in Germany.[28]

Personality

Noether was commonly regarded as a generous and caring woman, fiercely dedicated to her fields of study. Although she sometimes acted rudely to 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.[29] Van der Waerden later described her this way: "Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all."[30]

Her frugal lifestyle was at first a necessity of not receiving a salary; even after the university began paying her modestly in 1923, she lived 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.[31] Mostly unconcerned about appearance and manners, she focused on her studies to the exclusion of romance and fashion. Czech-American mathematician Olga Taussky-Todd described a luncheon wherein Noether, wholly engrossed by a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped it off from her dress, completely unperturbed".[32] Her appearance-conscious students cringed as she retrieved the handkerchief from her blouse and ignored her hair's growing disarray during a lecture. Two female pupils once approached her during a break in a two-hour class to express their concern, but were unable to break through the energetic mathematics discussion she was having with other students.[33]

Her lectures are described as enlightening but intense. She spoke quickly (to reflect the speed of her thoughts, many said), and demanded focused concentration from her students. Some pupils felt that she relied too much on spontaneous discussions; one wrote in a notebook with regard to a class which ended at 1:00 PM: "It's 12:50, thank God!"[34] Her most dedicated students, however, relished the enthusiasm with which she approached mathematics, especially since her lectures often built upon earlier work they had done together. She developed a close circle of colleagues and students, who thought along similar lines and frequently excluded those who did not. "Outsiders" who occasionally visited Noether's lectures usually spent only thirty minutes in the room before leaving in frustration or confusion. A regular student at one such instance was heard to remark: "The enemy has been defeated; he has cleared out."[35] Noether showed a devotion to the subject and her students that went beyond the regular school 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.[36] Later, after she had been dismissed by the Third Reich, she invited students into her home to discuss their future plans and mathematical concepts.[37]

1932

Noether and the Austrian mathematician Emil Artin were awarded the Ackermann–Teubner Memorial Award in 1932 for their contributions to mathematics. The prize came with a sum of 500 Reichsmarks (about US$120) and was seen as a long-overdue official recognition of her considerable work in the field. Her colleagues have expressed frustration at the fact that she was never elected to the Göttingen Gesellschaft der Wissenschaften (Academy of Sciences) and was never promoted to the position of Ordentlicher Professor.[38]

Noether's fiftieth birthday occurred in 1932 and her colleagues celebrated it with a typical mathematical style. Helmut Hasse dedicated an article to her in the Mathematische Annalen, wherein he confirmed her theory about the increased regularity of the laws governing noncommutative algebra compared to those of commutative. He also sent her a mathematical riddle, which she solved with great speed.[38]

In September of the same year, Noether delivered a major address (grosse Vorträge) at the International Congress of Mathematicians in Zürich, Switzerland. The conference was attended by 800 people, with 420 official participants. Her talk, on "Hyper-complex systems in their relations to commutative algebra and to number theory", was one of 21 major addresses at the congress. Notable participants of the 1932 ICM included Hermann Weyl, Edmund Landau, and Wolfgang Krull. Because her prominent speaking position was a recognition of her importance to the field of mathematics, the congress is sometimes described as the high point of her career.[39]

Expulsion

When Adolf Hitler became Chancellor of Germany in January 1933, Nazi activity around the country – including at the University of Göttingen – increased dramatically. The campus German Students Association led the charge against the "un-German Spirit", aided by a Privatdozent named Werner Weber, a former student of Noether. Anti-Semitic attitudes created a climate hostile to Jewish professors; one young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."[40] Several of Noether's colleagues, including Max Born and Richard Courant, had their positions revoked.[41]

In April 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 April 7, 1933, I hereby withdraw from you the right to teach at the University of Göttingen."[41] Noether accepted the decision calmly, providing support for others during the difficult time. Weyl wrote later: "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."[40] As usual, 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.[41]

Bryn Mawr

As dozens of newly-unemployed professors began searching for positions outside of Germany, their colleagues in the United States worked to provide assistance and opportunities. Einstein and Weyl were welcomed by Princeton University, while others worked to find the sponsor required for legal immigration. Noether was contacted by representatives of two schools, Bryn Mawr College and Oxford University; after a series of negotiations with the Rockefeller Foundation, a grant was approved and she took a position at Bryn Mawr starting in the winter of 1933–34.[42]

At Bryn Mawr, Noether met and befriended Anna Johnson Pell Wheeler, who had studied at Göttingen just before Noether arrived there. Another source of support at the college was Bryn Mawr President Marion Edwards Park, who enthusiastically invited mathematicians in the area to "see Dr. Noether in action!"[43] Noether and a small team of students worked quickly through van der Waerden's 1930 book Algebra I and parts of Erich Hecke's Theorie der algebraischen Zahlen (Theory of Algebraic Numbers, 1908).[44]

In 1934 Noether began lecturing at Princeton's Institute for Advanced Study, since – in her words – she was not welcome at the "men's university, where nothing female is admitted".[45] In addition to Abraham Flexner and Oswald Veblen, who had invited her, she worked with and supervised Abraham Adrian Albert and Harry Vandiver.[46] Her time in the US was pleasant, surrounded as she was by supportive colleagues and ensconced in her favorite subjects.[47] In the summer of 1934, she returned to Germany to see Artin and her brother Fritz before he left for Siberia. Although the universities had been cleared of many of her former colleagues, she was able to use the library as a "foreign scholar".[48]

Death

In April 1935, doctors discovered a tumor in Noether's pelvis. Because they were 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 the surgery. For three days she appeared to convalesce normally, and recovered quickly from a circulatory collapse on the fourth. On 14 April, she 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."[49]

Several days after Noether's death, her friends and associates at Bryn Mawr gathered at President Park's house, where a small memorial service took place. Hermann Weyl and Richard Brauer traveled from Princeton, and spoke with Wheeler and Taussky about their departed colleague. In the months which followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, and Alexandroff in paying respects. Her body was cremated, and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.[50]

Contributions to mathematics and physics

First and foremost, Noether is remembered as an algebraist, although her work had far-ranging consequences for theoretical physics and topology. She showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.[51] Her friend and colleague Hermann Weyl described her scholarly output in three epochs. In the first epoch (1908–1919), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Albert Gordan which she later characterized as Mist (manure) and Formelngestrüpp (a jungle of equations). Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. After moving to Göttingen in 1915, she produced her seminal work for physics, the two Noether's theorems. In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings.[52] In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields.[53]

Historical context

In the century from 1832 to Noether's death in 1935, the field of mathematics – specifically algebra – underwent a profound revolution, whose reverberations are still being felt.[54] Mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g., cubic, quartic and quintic equations, and on the related problem of constructing regular polygons using compass and straightedge. Beginning with Carl Friedrich Gauss' 1829 proof that prime numbers such as 5 can be factored in Gaussian integers, Évariste Galois' introduction of groups in 1832 and William Rowan Hamilton's discovery of quaternions in 1843, however, research turned to determining the properties of ever-more-abstract systems defined by ever-more-primitive rules. Noether's most important contributions were to the development of such abstract algebra, which have had numerous applications in mathematics and in real-world applications such as error-correcting codes used in transmitting and storing most digital information.

Although abstract algebra is a very technical field, some elements can be understood by analogy to the integers. The set of integers forms a structure known as a ring, because they can be added and multiplied and the result is always another integer. However, they do not form a field, because the result of dividing one integer by another integer might not itself be an integer. The multiples of any integer—say, the multiples of 3, namely ..., −6, −3, 0, 3, 6, 9,...—form a type of set called an ideal in the ring of integers. The addition and multiplication of integers is commutative, meaning that their result is independent of order: for any two numbers a and b, a + b = b + a and a × b = b × a. However, multiplication in other rings need not be commutative; some examples of non-commutative rings are formed by matrices or quaternions. The number of integers is infinite, but many rings are finite; for example, if addition and multiplication of integers are carried out modulo 3, then the ring has only three elements 0, 1 and 2. These correspond to the three residue classes associated with the ideal of 3 in the ring of integers.

It is important to realize that the definition of a "ring" is very general; the objects need not be integers and the two operations need not be customary addition and multiplication. For example, the objects might be computer data words and the operations some binary operations such as XOR. Therefore, many properties of the integers do not always pertain to more general rings. An important example is the fundamental theorem of arithmetic, which says that every positive integer can be factored uniquely into prime numbers; unique factorizations do not always exist in other rings. Much of Noether's work lay in determining what truths 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. For example, unique factorization domains are those commutative rings in which its members can be written uniquely as a product of prime elements. Noether did likewise for fields and modules; since fields are special cases of rings, they have additional properties that allow more truths to be derived about them. Modules are analogous to a vector space over a ring, in the same way that a normal vector is composed of three real-number components. Finally, there is a fundamental divide between commutative and non-commutative algebras; Noether was able to exploit their properties to derive truths unique to each type.

First epoch (1908–1919)

Invariant theory and elimination theory

Much of Noether's work in the first epoch was associated with invariant theory, beginning with her dissertation calculations under Gordan. Invariant theory is concerned with developing expressions that remain constant (invariant) under a group of transformations. As a simple example, if a rigid yardstick is rotated, the coordinates of its endpoints change, but its length L remains the same. In particular, some mathematicians studied the symmetric functions (particularly symmetric polynomials) that remain invariant under permutation of the roots of another function. Invariant theory was an active area of research in the later 19th 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.

Elimination theory is a closely related field concerned with eliminating a variable from a system of polynomial equations, usually by the method of resultants. For illustration, the system of equations can often be written in the form of a matrix M (missing the variable x) times a vector v (having only different powers of x) equaling a zero vector, M·v=0. Hence, the determinant of the matrix M must be zero, providing a new equation in which the variable x has been eliminated.

Galois theory

Galois theory is related to invariant theory, and concerns transformations of number fields that permute the roots of an equation. Consider an polynomial equation of a variable x of degree n, in which the coefficients are drawn from some "ground" field, which might be the field of real numbers or of rational numberss or of the integers modulo 7. The n root (mathematics) of such an equation need not lie within the ground field; for example, the roots of a real polynomial equation may be complex numbers, which is an extension of the real number field by adjoining the imaginary number i. More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field.

The Galois group of an equation is the set of one-to-one transformations (in technical language, automorphisms) of the extension field that leave the ground field unchanged. For example, the Galois group for a real polynomial might be a warping of the complex plane that leaves the axis of real numbers unchanged. Since the Galois group doesn't change the ground field, it leaves the coefficients of the polynomial unchanged, and thus, it can at most permute the n roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory, which draws a one-to-one parallel between the subgroups of the Galois group and subextensions of ground field.

In 1918, Noether published a seminal paper on the inverse Galois problem.[55] 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 is always 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 Sn acting on the field k(x1,...,xn) is always a pure transcendental extension of the field k. (She first mentioned this problem in a 1913 paper,[56] where she attributed the problem to her colleague Fischer.) She showed this was true for n=2, 3, or 4. In 1969, R. G. Swan found a counter-example to Noether's problem, with n=47 and G a cyclic group of order 47;[57], (though this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem is still being researched actively.

Physics

Main articles: Noether's theorem, Conservation law, and Constant of motion

Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding the general theory of relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, due to the fact that gravitational energy could itself gravitate. Emmy Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, with her two Noether's theorems, which she proved in 1915 but published only in 1918.[58] She solved the problem not only for general relativity, but determined the conserved quantities for every system of physical laws that possesses some continuous symmetry. Upon receiving her work, Einstein wrote to Hilbert,[59] "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."

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.[60] 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 has the same outcome regardless of place or time (working the same, say, in Cleveland on Tuesday and Samaria on Wednesday), 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.

Noether's two theorems have become a fundamental tool of modern theoretical physics, both because of the insight they give into conservation laws, and also as a practical calculation tool.[3] They allow researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, they allow researchers to consider whole classes of hypothetical physical laws to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether's theorems, the physical laws that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria. The converse of Noether's theorem is not always true; not every conservation law corresponds to a continuous symmetry.[60]

Second epoch (1920–1926)

Commutative rings, ideals and modules

Her 1921 paper Idealtheorie in Ringbereichen[61] is the foundation of general commutative ring theory, and gives the first general definition of a commutative ring. Before this 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.[62] A Noetherian module is a module that satisfies the ascending chain condition on submodules, where the submodules are partially ordered by inclusion. (A Noetherian topological space is one that satisfies a similar ascending chain condition on open subsets, for example the spectrum of a Noetherian ring, though Noether never worked on these.) A major result in this paper is the Lasker–Noether theorem in commutative algebra, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings.

Noether's 1927 work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern[63] characterized the rings in which the ideals have unique factorization into prime ideals as the Dedekind domains: integral domains that are Noetherian, 0 or 1-dimensional, and integrally closed in their quotient fields. This paper also contains what are now called the isomorphism theorems, which describe some fundamental natural isomorphisms, and some other basic results on Noetherian and Artinian modules.

Return to invariant theory

In her 1926 paper[64] she extended Hilbert's theorem on the finite generation of rings of invariants of finite groups from the characteristic 0 case to characteristic p>0. (Noether's result was later extended by Haboush to all reductive groups in his proof of Mumford's conjecture.) In this paper Noether also introduced the Noether normalization lemma, showing that a finitely generated domain A over a field k has a transcendence basis x1,...xn such that A is integral over k[x₁,...x_n].

Third epoch (1927–1935)

Hypercomplex numbers and representation theory

Much work on hypercomplex numbers and the representations of groups was carried out in the 19th and early 20th centuries; but this disparate work was united by Noether, who gave the first general representation theory of groups and algebras.[65] Briefly, Noether subsumed the structure theory of abstract algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings that satisfy certain finiteness conditions. This single work had a profound impact on the development of modern algebra.[54]

Non-commutative algebra

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

A seminal paper by Noether, Helmut Hasse and Richard Brauer pertains to division algebras,[66] which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a division algebra over a number field splits locally everywhere then it splits globally, and from this deduced their Hauptsatz ("main theorem"): Every finite dimensional central division algebra over an algebraic number field F is a cyclic algebra over F. (A cyclic algebra of dimension n2 over a field F is one that contains a cyclic sub-extension of dimension n.) A subsequent paper by Noether[67] showed, as a special case of a more general theorem, that all maximal subfields of a division algebra D are splitting fields. This paper also contains the Skolem–Noether theorem which states that any two embeddings of an extension of a field k into a finite dimensional central simple algebra over k are conjugate. The Brauer-Noether theorem [68] gives a characterization of the splitting fields of a central division algebra over a field.

Legacy

Noether's influence on the fields of algebra and physics was evident even before her death. In a letter to the New York Times just after her death, Albert Einstein wrote:

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.... Emmy Noether, who, in spite of the efforts of the great Göttingen mathematician, Hilbert, never reached the academic standing due her in her own country, none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators.[1]

Her colleagues Pavel Alexandrov and Hermann Weyl agreed with Einstein that she was the greatest woman ever to work in the field.[2] In a 1964 World's Fair exhibit entitled "Men of Modern Mathematics", Noether was the only female represented.[69]

Evidence of her impact is reflected in the ubiquitous presence of her name in the worlds of science and mathematics. The University of Siegen houses its mathematics and physics buildings on the Emmy Noether Campus.[70] The Nöther crater, located on the far side of the Moon is named for her, as is the 7001 Noether asteroid.[71][72] Each year the Association for Women in Mathematics presents the Noether Lecture to honor women in mathematics. A 2005 pamphlet about the series notes: "Emmy Noether was one of the great mathematicians of her time, someone who worked and struggled for what she loved and believed in. Her life and work remain a tremendous inspiration."[73]

Notes

1. ^ a b Einstein, Albert. "Professor Einstein Writes in Appreciation of a Fellow-Mathematician". 5 May 1935. Online at the MacTutor History of Mathematics archive. Retrieved on 13 April 2008.
2. ^ a b Osen 1974, p. 152; Alexandrov 1981, p. 100.
3. ^ a b Ne'eman Y (1999). "The Impact of Emmy Noether's Theorems on XX1st Century Physics", in M Teicher: The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, Volume 12, pp. 83–101.
4. ^ a b Lederman & Hill 2004, p. 73.
5. ^ Kimberling 1981, pp. 3–5; Osen 1974, p. 142; Lederman & Hill 2004, pp. 70–71; Dick 1981, pp. 7–9.
6. ^ Dick 1981, pp. 9–10.
7. ^ Dick 1981, pp. 10–11; Osen 1974, p. 142.
8. ^ Dick 1981, pp. 25, 45; Kimberling, p. 5.
9. ^ Quoted in Kimberling (1981, p. 10).
10. ^ Dick 1981, pp. 11–12; Kimberling 1981, pp. 8–10; Lederman & Hill 2004, p. 71.
11. ^ Kimberling 1981, pp. 10–11; Dick 1981, pp. 13–17. Lederman & Hill (2004, p. 71) write that she completed her doctorate at Göttingen, but this appears to be an error.
12. ^ Kimberling 1981, pp. 11–12; Dick 1981, pp. 18–24; Osen 1974, p. 143.
13. ^ a b Kimberling 1981, p. 14; Dick 1981, p. 32; Osen 1974, pp. 144–145; Lederman & Hill 2004, p. 72.
14. ^ Dick 1981, pp. 24–26.
15. ^ Osen 1974, pp. 144–145; Lederman & Hill 2004, p. 72.
16. ^ Kimberling 1981, p. 14–18; Osen 1974, p. 145; Dick 1981, pp. 33–34.
17. ^ Kimberling 1981, p. 18.
18. ^ Kimberling 1981, p. 18; Dick 1981, pp. 44–45; Osen 1974, pp. 145–146.
19. ^ Dick 1981, p. 51.
20. ^ Dick 1981, pp. 53–57.
21. ^ Van der Waerden 1935, p. 100.
22. ^ Dick 1981, pp. 57–58; Kimberling 1981, p. 19; Lederman & Hill 2004, p. 74.
23. ^ Lederman & Hill 2004, p. 74; Osen 1974, p. 148.
24. ^ Kimberling 1981, pp. 24–25; Dick 1981, pp. 61–63.
25. ^ Alexandrov 1981, pp. 100, 107.
26. ^ Dick 1981, pp. 63–64; Kimberling 1981, p. 26; Alexandrov 1981, pp. 108–110.
27. ^ a b Alexandrov 1981, pp. 106–109.
28. ^ Osen 1974, p. 150; Dick 1981, pp. 82–83.
29. ^ Dick 1981, pp. 37–49.
30. ^ Van der Waerden 1935, p. 98.
31. ^ Dick 1981, pp. 46–48.
32. ^ Taussky 1981, p. 80
33. ^ Dick 1981, pp. 40–41.
34. ^ Mac Lane 1981, p. 77; Dick 1981, p. 37.
35. ^ Dick 1981, pp. 38–41.
36. ^ Mac Lane 1981, p. 71
37. ^ Dick 1981, p. 76
38. ^ a b Dick 1981, pp. 72–73; Kimberling 1981, pp. 26–27.
39. ^ Kimberling 1981, p. 26–27; Dick 1981, pp. 74–75.
40. ^ a b Kimberling 1981, p. 29.
41. ^ a b c Dick 1981, pp. 75–76; Kimberling 1981, pp. 28–29.
42. ^ Dick 1981, pp. 78–79; Kimberling 1981, pp. 30–31.
43. ^ Kimberling 1981, pp. 32–33; Dick 1981, p. 80. (Exclamation point in the original.)
44. ^ Dick 1981, pp. 80–81.
45. ^ Dick 1981, p. 81.
46. ^ Dick 1981, pp. 81–82.
47. ^ Osen 1974, p. 151; Dick 1981, p. 83.
48. ^ Dick 1981, p. 82; Kimberling 1981, p. 34.
49. ^ Kimberling 1981, pp. 37–38.
50. ^ Kimberling 1981, p. 39.
51. ^ Osen 1974, pp. 148–149; Kimberling 1981, p. 11–12.
52. ^ Gilmer 1981, p. 131.
53. ^ Kimberling 1981, pp. 10–23.
54. ^ a b Van der Waerden 1985.
55. ^ Noether 1918.
56. ^ Noether 1913.
57. ^ Swan 1969.
58. ^ Noether 1918b
59. ^ Kimberling, p. 13.
60. ^ a b Lederman & Hill 2004, pp. 97–116.
61. ^ Noether 1921.
62. ^ Gilmer 1981, p. 133.
63. ^ Noether 1927.
64. ^ Noether 1926.
65. ^ Noether 1929.
66. ^ Brauer, Hasse & Noether 1932.
67. ^ Noether 1933.
68. ^ Brauer & Noether 1927
69. ^ Duchin, Moon. "The Sexual Politics of Genius". December 2004. University of Chicago. Retrieved on 13 April 2008.
70. ^ "Emmy-Noether-Campus". Universität Siegen. Retrieved on 13 April 2008.
71. ^ Schmadel 2003, p. 570.
72. ^ Blue, Jennifer. Gazetteer of Planetary Nomenclature. USGS. 25 July 2007. Retrieved on 13 April 2008.
73. ^ "Introduction". Profiles of Women in Mathematics: The Emmy Noether Lectures. Association for Women in Mathematics. 2005. Retrieved on 13 April 2008.

References

* Alexandrov, Pavel S. (1981), "In Memory of Emmy Noether", in James W. Brewer and Martha K. Smith, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 99–111, ISBN 0-8247-1550-0 .

* Blue, Meredith (2001), Galois Theory and Noether's Problem, Thirty-Fourth Annual Meeting: Florida Section of The Mathematical Association of America, <http://mcc1.mccfl.edu/fl_maa/proceedings/2001/blue.pdf> .

* Brauer, R. & Noether, Emmy (1927), "Über minimale Zerfällungskörper irreduzibler Darstellungen", Sitz. Ber. d. Preuss. Akad. d. Wiss.: 221-228

* Brauer, R.; Hasse, H. & Noether, E. (1932), "Beweis eines Hauptsatzes in der Theorie der Algebren", Journal für Math. 167: 399–404

* Dick, Auguste (1981), Emmy Noether: 1882–1935, Boston: Birkhäuser, ISBN 3-7643-3019-8 . Trans. H.I. Blocher.

* Gilmer, Robert (1981), "Commutative Ring Theory", in James W. Brewer and Martha K. Smith, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 131–143, ISBN 0-8247-1550-0 .

* Kimberling, Clark (1981), "Emmy Noether and Her Influence", in James W. Brewer and Martha K. Smith, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 3–61, ISBN 0-8247-1550-0 .

* Lederman, Leon M. & Hill, Christopher T. (2004), Symmetry and the Beautiful Universe, Amherst: Prometheus Books, ISBN 1-59102-242-8 .

* Mac Lane, Saunders (1981), "Mathematics at the University of Göttingen 1831–1933", in James W. Brewer and Martha K. Smith, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 65–78, ISBN 0-8247-1550-0 .

* Noether, Emmy (1913), "Rationale Funkionenkorper", J. Ber. d. DMV 22: 316–319 .

* Noether, Emmy (1918), "Gleichungen mit vorgeschriebener Gruppe", Mathematische Annalen 78: 221–229, DOI 10.1007/BF01457099 .

* Noether, Emmy (1918b), "Invariante Variationsprobleme", Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse 1918: 235–257, <http://arxiv.org/PS_cache/physics/pdf/0503/0503066v1.pdf> .

* Noether, Emmy (1921), "Idealtheorie in Ringbereichen", Mathematische Annalen 83 (1), ISSN 0025-5831, <http://www.springerlink.com/content/m3457w8h62475473/fulltext.pdf> .

* Noether, Emmy (1926), "Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p", Nachr. Ges. Wiss. Göttingen: 28–35, <http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=63971> .

* Noether, Emmy (1927), "Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern", Mathematische Annalen 96 (1): 26–61, ISSN 0025-5831, <http://www.springerlink.com/content/v3t6331n8w244275/fulltext.pdf> .

* Noether, Emmy (1929), "Hyperkomplexe Grössen und Darstellungstheorie", Mathematische Annalen 30: 641–692, DOI 10.1007/BF01187794 .

* Noether, Emmy (1933), "Nichtkommutative Algebren", Mathematische Zeitschrift 37: 514–541, DOI 10.1007/BF01474591 .

* Noether, Emmy (1983), Jacobson, Nathan, ed., Gesammelte Abhandlungen (Collected papers), Berlin-New York: Springer-Verlag, pp. viii+777, MR0703862, ISBN 3-540-11504-8 .

* Osen, Lynn M. (1974), "Emmy (Amalie) Noether", Women in Mathematics, MIT Press, pp. 141–152, ISBN 0-262-15014-X .

* Schmadel, Lutz D. (2003), Dictionary of Minor Planet Names (5th revised and enlarged ed.), Berlin: Springer-Verlag, ISBN 3-540-00238-3 .

* Swan, R. G. (1969), "Invariant rational functions and a problem of Steenrod", Inventiones Mathematicae 7: 148–158, DOI 10.1007/BF01389798

* Taussky, Olga (1981), "My Personal Recollections of Emmy Noether", in James W. Brewer and Martha K. Smith, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 79–92, ISBN 0-8247-1550-0 .

* Van der Waerden, B.L. (1935), "Nachruf auf Emmy Noether (Obituary of Emmy Noether)", Mathematische Annalen 111: 469–474, DOI 10.1007/BF01472233 . Reprinted in Dick 1981.

* Van der Waerden, B.L. (1985), A History of Algebra, from al-Khwārizmi to Emmy Noether, Berlin: Springer-Verlag, ISBN 0-387-13610-X .