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Emmy Noether

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Emmy Noether, c. 1905

Amalie Emmy Noether, IPA: [ˈnøːtɐ], (March 23 1882 – April 14 1935) was a German Jewish mathematician who is known for her seminal contributions to abstract algebra. Often described as the most important woman in the history of mathematics,[1][2] she revolutionized the theories of rings, fields, and algebras. She is also known for her contributions to modern theoretical physics; the two Noether's theorems explain the connections between symmetry and conservation laws, and are essential tools of research.[3]

Born in the Bavarian town of Erlangen to the noted mathematician Max Noether and his wife, Emmy originally planned to teach French and English after passing the required examinations. She did not pursue languages, however, and studied 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 in 1919, paving the way for her to obtain the rank of Privatdozent. She remained at Göttingen until 1933, where she was a leading member of a world-renowned center of mathematics; her students were sometimes called the "Noether boys". In 1924 Dutch mathematician Bartel Leendert van der Waerden joined her circle of students and quickly became her leading expositor; her work was the foundation for the second volume of his influential 1931 textbook Moderne Algebra. By the time she delivered a major address at the 1932 International Congress of Mathematicians in Zürich, Switzerland, her algebraic acumen was recognized around the world. The following year, Germany's Nazi government had her fired 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 "epochs". In the first (1908–19), she made valuable contributions to invariant theory, most notably 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 (1920–26), Noether studied commutative rings, working out the theory of ideals in such rings. Her paper Idealtheorie in Ringbereichen (1921) is considered a classic work among mathematicians. In it, she gave the first definition of a commutative ring and used ascending chain conditions to prove an analog of the fundamental theorem of arithmetic. Rings and modules that satisfy the ascending chain condition are called Noetherian in her honor. In the third epoch (1927–35), she worked on noncommutative algebras, as well as uniting hypercomplex numbers and the representation theory of groups with the modules and ideals of rings. She also formulated a theory of modules and ideals in rings that satisfy certain finiteness conditions. Her papers Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract composition of ideal theory in algebraic number and function fields, 1927), Hyperkomplexe Größen und Darstellungstheorie (Hypercomplex quantities and representation theory, 1929) and Beweis eines Hauptsatzes in der Theorie der Algebren (Proof of a main theorem in the theory of algebras, 1932) laid important groundwork for modern abstract algebra. Noether was 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

Noether grew up in the Bavarian city of Erlangen, depicted here in a 1916 postcard.

Emmy's father, Max Noether, was descended from a family of Jewish wholesale traders in Germany. He had been paralyzed by poliomyelitis at the age of 14; even after regaining mobility, he was handicapped in one leg. Largely self-taught, 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. His most famous results are the Brill–Noether theorem and the residue or AF+BG theorem; he is also known, however, for several other theorems.

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. Known for being clever and friendly, Emmy was near-sighted and talked during childhood with a minor lisp. A family friend recounted a story years later about young Emmy quickly solving a brain teaser at a children's party, showing logical acumen as a youth.[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. After studying in Munich, he made a reputation for himself in the field of applied mathematics. Her eldest brother, Alfred, was born in 1883, received a doctorate in chemistry from Erlangen in 1909, and died nine years later. The youngest, Gustav Robert, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928.[8]

Paul Albert Gordan supervised Noether's doctoral dissertation on invariants of biquadratic forms.

University of Erlangen

Emmy 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 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 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 at the University of Göttingen, attending lectures from astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert. Soon after, 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 supervision 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 Fischer by mailing commentary written on postcards.[12]

University of Göttingen

File:Hilbert1912.jpg
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 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 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]

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. Around the same time, Noether's father retired and her brother joined the German 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, and Noether would provide "assistance". However, soon after arriving, 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] American physicists Leon M. 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 Noether to proceed with the Habilitation, a process to obtain the rank of Privatdozent. Her oral examination was 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. This was an unpaid "extraordinary" professorship, not the higher "ordinary" professorship, which was a civil-service position. Although it recognized the importance of her work, the position still provided no salary; not until she was appointed to the special position of Lehrauftrag für Algebra one year later was she paid for her lectures.[16]

In 1920 Noether collaborated with a colleague named W. Schmeidler on a paper about the theory of ideals, in which they defined left and right ideals. The following year she published a landmark paper called Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regard to ideals. Canadian mathematician Irving Kaplansky has called this work "revolutionary";[17] it gave rise to the term "Noetherian ring".[18]

The mathematics department at the University of Göttingen allowed Noether's Habilitation in 1919, four years after she began lecturing at the school

Soon afterwards, she began supervising doctoral students, including Grete Hermann, who later spoke reverently of her "dissertation-mother".[19] Noether also supervised 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]

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.[21] A colleague 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."[22]

Her frugal lifestyle was at first a necessity of not receiving a salary; however, 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.[23] 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 in which 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".[24] Her 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 were unable to break through the energetic mathematics discussion she was having with other students.[25]

Her lectures are described as enlightening but intense. She spoke quickly (reflecting the speed of her thoughts, many said) and demanded great concentration from her students. Students who disliked her style often felt alienated; one wrote in a notebook with regard to a class that ended at 1:00 pm: "It's 12:50, thank God!"[26] Some pupils felt that she relied too much on spontaneous discussions. Her most dedicated students, however, relished the enthusiasm with which she approached mathematics, especially since her lectures often built on earlier work they had done together. She developed a close circle of colleagues and students who thought along similar lines and that typically excluded those who did not. "Outsiders" who occasionally visited Noether's lectures usually spent only 30 minutes in the room before leaving in frustration or confusion. A regular student at one such instance said: "The enemy has been defeated; he has cleared out."[27] 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.[28] Later, after she had been dismissed by the Third Reich, she invited students into her home to discuss their future plans and mathematical concepts.[29]

In 1924 the young Dutch mathematician Bartel Leendert van der Waerden arrived at the University of Göttingen. He began working immediately with Noether, who provided invaluable methods of abstract conceptualization. He said later that her originality was "absolute beyond comparison".[30] In 1931 he published Moderne Algebra, a central text in the field; its second volume borrows heavily from Noether's work. Although she did not seek recognition, 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".[31] She sometimes allowed her colleagues and students to receive credit for her ideas, helping them develop their careers at the expense of her own.[32]

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 able to help him secure only a scholarship from the Rockefeller Foundation.[33] They met regularly and enjoyed discussions about the intersections of algebra and topology. In his 1935 memorial address, Alexandrov named her "the greatest woman mathematician of all time".[34]

Moscow

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Noether taught at the University of Moscow during the winter of 1928–1929.

In the winter of 1928–1929, 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 topologists Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory.[35]

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".[36]

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.[36] Her brother Fritz, meanwhile, took a position at the Research Institute for Mathematics and Mechanics in Tomsk, Siberia after losing his job in Germany.[37]

Recognition

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 and was seen as a long-overdue official recognition of her considerable work in the field. Her colleagues have since 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 visited Zürich in 1932 to deliver a major address at the International Congress of Mathematicians.

Noether's fiftieth birthday occurred in 1932 and her colleagues celebrated it in typical mathematical style. Helmut Hasse dedicated an article to her in the Mathematische Annalen, wherein he confirmed her suspicion that some aspects of noncommutative algebra are simpler than those of commutative algebra, by proving a noncommutative reciprocity law.[39] This pleased her immensely. He also sent her a mathematical riddle, the "mμν-riddle of syllables". She solved it immediately, but the riddle itself has been lost.[38]

In September of the same year, Noether delivered a major address (großer Vortrag) at the International Congress of Mathematicians in Zürich, Switzerland. The conference was attended by 800 people, with 420 officially participating. Notable participants included Hermann Weyl, Edmund Landau, and Wolfgang Krull. 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. 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.[40]

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. Antisemitic attitudes created a climate hostile to Jewish professors; one young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."[41] Several of Noether's colleagues, including Max Born and Richard Courant, had their positions revoked.[42]

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."[42] 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."[41] 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.[42]

Bryn Mawr

Bryn Mawr College provided a welcoming home for Noether during the last two years of her life.

As dozens of newly-unemployed professors began searching for positions outside of Germany, their colleagues in the United States 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 the University of Oxford. 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 1933–1934.[43]

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!"[44] 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).[45]

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".[46] In addition to Abraham Flexner and Oswald Veblen, who had invited her, she worked with and supervised Abraham Adrian Albert and Harry Vandiver.[47] Her time in the United States was pleasant, surrounded as she was by supportive colleagues and ensconced in her favorite subjects.[48] In the summer of 1934, she returned to Germany briefly 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".[49]

Noether's remains are buried under the walkway surrounding the cloisters of Bryn Mawr's M. Carey Thomas Library.

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".[50] 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 (42.78 °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."[50]

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 Alexandrov 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.[51]

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.[52] 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 (crap) 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.[53] In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields.[54]

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.[55] 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-universal rules. Noether's most important contributions to mathematics were to the development of this new field, abstract algebra,[56] which has had numerous applications in mathematics and in real-world applications such as coding theory; for example, the Reed-Solomon code, used to store the data on compact discs and DVDs, makes it possible to recover data even in the presence of small scratches on the disc.

Despite the generality of abstract algebra, some elements may be understood by analogy to the integers. Any two integers can be added or multiplied, and the result of these operations is always another integer. Much of Noether's work was with abstract systems of addition and multiplication known as rings. Rings are sets whose elements can be treated like numbers: They can be added and multiplied, and the operations of addition and multiplication obey rules similar to the addition and multiplication of integers. For example, addition of integers is commutative, meaning that a + b (in that order) is the same as b + a. Similarly, the sum of two elements of a ring is independent of their order. However, the product in a ring is not assumed to be commutative, meaning that a × b might be different from b × a. Examples of noncommutative rings include matrices and quaternions. It is also not always possible to divide two elements of a ring. This is true even in the integers: 1 and 3 are both integers, but 1/3 is not. Instead it is only possible to divide two numbers when their quotient has no remainder. For example, a number is evenly divisible by 3 exactly when it is a multiple of 3 — that is, when it is one of .., −6, −3, 0, 3, 6, 9,.... The set of all multiples of 3 form an ideal of the ring of integers. Questions about divisibility in the integers can be translated to other rings by rephrasing them using ideals. The number of integers is infinite, but many rings are finite; for example, the hours displayed on a clock are the elements of a ring. The addition and multiplication in this ring are the addition and multiplication of integers carried out modulo 12, meaning that two elements are considered equal if their difference is divisible by 12. This ring has 12 elements: 0, 1, 2, 3, and so on up to 11. 0 is usually written 12 on a clock, but they represent the same element in this ring because their difference, which is 12, is divisible by 12. The elements of this ring are the twelve residue classes associated with the ideal of 12 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, where addition is chosen to be exclusive or and multiplication is chosen to be logical conjunction. 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, but Noether found a unique factorization theorem, now called 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.

First epoch (1908–1919)

Noether sometimes used postcards to discuss abstract algebra with her colleague Ernst Sigismund Fischer. This card was postmarked 10 April 1915.

Algebraic invariant theory

Much of Noether's work in the first epoch 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 (x, y, z) of its endpoints change, but its length L – given by the formula L2 = Δx2 + Δy2 + Δz2 – remains the same. In particular, some mathematicians studied the symmetric functions (particularly the 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.

Noether's dissertation was an extension of the work of her advisor, Paul Albert Gordan, from two variables to three variables. Gordan's chief contribution to mathematics was his proof that invariant homogeneous polynomials in two variables, x and y, can be produced from a finite number of invariant polynomials, called generators, by performing a finite number of addition and multiplication operations. He proved this by finding formulas for all of the invariants, but he was unable to carry out this constructive approach for invariants in three or more variables. In the 1890s David Hilbert proved a similar statement for the invariant polynomials for a wide variety of group actions, but his proof only guaranteed that this could be done and did not find formulas for the generators. Gordan initially dismissed Hilbert's proof as "theology". Noether herself accepted Hilbert's proof, but noted that her constructive approach made it possible to study the relationships among the invariants.

Galois theory

Galois theory is related to invariant theory and concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable x of degree n, 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 x which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. 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. However, 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, i and −i, where i is the imaginary unit, that is, i2 = −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.

The Galois group of a polynomial is the set of all ways of transforming the splitting field while preserving the ground field and the roots of the polynomial. (In mathematical jargon, these transformations are called automorphisms.) The Galois group of x2 + 1 consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends i to −i. Since the Galois group doesn't change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. However, each root can move to another root, 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.

In 1918, Noether published a seminal paper on the inverse Galois problem.[57] 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,[58] 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[59] (though this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem is still unsolved.

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 did not publish until 1918.[60] 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: "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."[61]

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.[62] 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 facilitate 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 theorems provide a test for theoretical explanations of the phenomenon: If the theory has a symmetry, then Noether's theorem guarantees that the theory has conserved quantity, and for the theory to be correct, this conservation must be observable in experiments. The converse of Noether's theorem is not always true; not every conservation law corresponds to a continuous symmetry.[62]

Second epoch (1920–1926)

Commutative rings, ideals, and modules

Her 1921 paper Idealtheorie in Ringbereichen (Ideal theory in domains)[63] is the foundation of general commutative ring theory, and gives one of the first general definitions of a commutative ring.[64] 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.[64] 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, e.g., the spectrum of a Noetherian ring; however, Noether never worked on these.) A major result in this 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 allows to write every positive integer as a product of prime numbers in a unique way.

Noether's 1927 work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Composition of Ideal Theory in Algebraic Number and Function Fields)[65] 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.

Elimination theory

In 1923–1924, Noether applied her ideal theory to elimination theory—in a formulation that she attributed to her student, Kurt Hentzelt[66]—showing that fundamental theorems about the factorization of polynomials could be carried over directly.[67] 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, 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 the 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.

Return to invariant theory

In her 1926 paper[68] she extended Hilbert's theorem on the finite generation of rings of invariants of finite groups. Hilbert's original proof was for rings which contained the rational numbers. Noether extended this to all Noetherian rings containing a field. (Noether's result was later extended by William 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, ..., xn].

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 19th and early 20th centuries, but remained disparate. Noether united the results and gave the first general representation theory of groups and algebras.[69] 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 certain finiteness conditions. This single work had a profound impact on the development of modern algebra.[55]

Noncommutative 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.[70]

A seminal paper by Noether, Helmut Hasse and Richard Brauer pertains to division algebras,[71] 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"): Every finite dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension. These theorems allow one to classify all finite dimensional central division algebras over a given number field. A subsequent paper by Noether[72] 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[73] gives a characterization of the splitting fields of a central division algebra over a field.

Legacy

The Emmy Noether Campus at the University of Siegen is home to the mathematics and physics departments. Image by Bob Ionescu.

Noether's contributions to the fields of algebra and physics have been hailed around the globe. 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] Historical representations of mathematics history confirm this: In a 1964 World's Fair exhibit entitled "Men of Modern Mathematics", Noether was the only female represented.[74]

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.[75] The Nöther crater, located on the far side of the Moon, is named for her, as is the 7001 Noether asteroid.[76][77] 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."[78]

See also

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, Yuval. "The Impact of Emmy Noether's Theorems on XX1st Century Physics". The Heritage of Emmy Noether. Ed. M. Teicher. Israel Mathematical Conference Proceedings. Bar-Ilan University/American Mathematical Society/Oxford University Press, 1999. OCLC 223099225. ISBN 978-0198510451. 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. ^ Dick 1981, pp. 37–49.
  22. ^ Van der Waerden 1935, p. 98.
  23. ^ Dick 1981, pp. 46–48.
  24. ^ Taussky 1981, p. 80
  25. ^ Dick 1981, pp. 40–41.
  26. ^ Mac Lane 1981, p. 77; Dick 1981, p. 37.
  27. ^ Dick 1981, pp. 38–41.
  28. ^ Mac Lane 1981, p. 71
  29. ^ Dick 1981, p. 76
  30. ^ Van der Waerden 1935, p. 100.
  31. ^ Dick 1981, pp. 57–58; Kimberling 1981, p. 19; Lederman & Hill 2004, p. 74.
  32. ^ Lederman & Hill 2004, p. 74; Osen 1974, p. 148.
  33. ^ Kimberling 1981, pp. 24–25; Dick 1981, pp. 61–63.
  34. ^ Alexandrov 1981, pp. 100, 107.
  35. ^ Dick 1981, pp. 63–64; Kimberling 1981, p. 26; Alexandrov 1981, pp. 108–110.
  36. ^ a b Alexandrov 1981, pp. 106–109.
  37. ^ Osen 1974, p. 150; Dick 1981, pp. 82–83.
  38. ^ a b Dick 1981, pp. 72–73; Kimberling 1981, pp. 26–27.
  39. ^ Hasse 1933
  40. ^ Kimberling 1981, p. 26–27; Dick 1981, pp. 74–75.
  41. ^ a b Kimberling 1981, p. 29.
  42. ^ a b c Dick 1981, pp. 75–76; Kimberling 1981, pp. 28–29.
  43. ^ Dick 1981, pp. 78–79; Kimberling 1981, pp. 30–31.
  44. ^ Kimberling 1981, pp. 32–33; Dick 1981, p. 80. (Exclamation point in the original.)
  45. ^ Dick 1981, pp. 80–81.
  46. ^ Dick 1981, p. 81.
  47. ^ Dick 1981, pp. 81–82.
  48. ^ Osen 1974, p. 151; Dick 1981, p. 83.
  49. ^ Dick 1981, p. 82; Kimberling 1981, p. 34.
  50. ^ a b Kimberling 1981, pp. 37–38.
  51. ^ Kimberling 1981, p. 39.
  52. ^ Osen 1974, pp. 148–149; Kimberling 1981, p. 11–12.
  53. ^ Gilmer 1981, p. 131.
  54. ^ Kimberling 1981, pp. 10–23.
  55. ^ a b Van der Waerden 1985.
  56. ^ Noether GE (1987), Grinstein LS, Campbell PJ (ed.), Women of Mathematics, New York: Greenwood press, p. 168, ISBN 0-313-24849-4
    American Council of Learned Societies (1991), Biographical Dictionary of Mathematicians: Reference Biographies from the Dictionary of Scientific Biography, vol. vol. 4, New York: Charles Scribner's Sons, p. 1870, ISBN 0-684-19291-8 {{citation}}: |volume= has extra text (help)
  57. ^ Noether 1918.
  58. ^ Noether 1913.
  59. ^ Swan 1969.
  60. ^ Noether 1918b
  61. ^ Kimberling 1981, p. 13.
  62. ^ a b Lederman & Hill 2004, pp. 97–116.
  63. ^ Noether 1921 harvnb error: multiple targets (2×): CITEREFNoether1921 (help).
  64. ^ a b Gilmer 1981, p. 133.
  65. ^ Noether 1927.
  66. ^ Noether, Emmy (1921), "Über eine Arbeit des im Kriege gefallenen K. Hentzelt zur Eliminationstheorie", Jahresbericht der Deutschen Mathematiker-Vereinigung, 30 (Abt. 2): 101
  67. ^ Noether, Emmy (1923), "Zur Theorie der Polynomideale und Resultanten", Mathematische Annalen, 88: 53–79
    Noether, Emmy (1923), "Eliminationstheorie und allgemeine Idealtheorie", Mathematische Annalen, 90: 229–261
    Noether, Emmy (1923), "Eliminationstheorie und Idealtheorie", Jahresbericht der Deutschen Mathematiker-Vereinigung, 33: 116–120
  68. ^ Noether 1926.
  69. ^ Noether 1929.
  70. ^ Lam 1981, pp. 152–153.
  71. ^ Brauer, Hasse & Noether 1932.
  72. ^ Noether 1933.
  73. ^ Brauer & Noether 1927
  74. ^ Duchin, Moon. "The Sexual Politics of Genius". December 2004. University of Chicago. Retrieved on 13 April 2008.
  75. ^ "Emmy-Noether-Campus". Universität Siegen. Retrieved on 13 April 2008.
  76. ^ Schmadel 2003, p. 570.
  77. ^ Blue, Jennifer. Gazetteer of Planetary Nomenclature. USGS. 25 July 2007. Retrieved on 13 April 2008.
  78. ^ "Introduction". Profiles of Women in Mathematics: The Emmy Noether Lectures. Association for Women in Mathematics. 2005. Retrieved on 13 April 2008.

References

Selected works by Emmy Noether (all in German)

Main article: List of publications by Emmy Noether

Secondary sources

  • Alexandrov, Pavel S. (1981), "In Memory of Emmy Noether", in James W. Brewer and Martha K. Smith (ed.), 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 (PDF), Thirty-Fourth Annual Meeting: Florida Section of The Mathematical Association of America.
  • Byers, Nina (2006), "Emmy Noether", in Nina Byers and Gary Williams (ed.), Out of the Shadows: Contributions of 20th Century Women to Physics, Cambridge: Cambridge University Press, ISBN 0-5218-2197-5.
  • 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 (ed.), Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 131–143, ISBN 0-8247-1550-0.
  • Hasse, Helmut (1933), "Die Struktur der R. Brauerschen Algebrenklassengruppe über einem algebraischen Zahlkörper", Mathematische Annalen, 107: 731–760. (German)
  • Kimberling, Clark (1981), "Emmy Noether and Her Influence", in James W. Brewer and Martha K. Smith (ed.), Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 3–61, ISBN 0-8247-1550-0.
  • Lam, T. Y. (1981), "Representation Theory", in James W. Brewer and Martha K. Smith (ed.), Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 145–156, 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 (ed.), Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc., pp. 65–78, ISBN 0-8247-1550-0.
  • 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 (ed.), 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. (German)
  • Van der Waerden, B.L. (1985), A History of Algebra: from al-Khwārizmī to Emmy Noether, Berlin: Springer-Verlag, ISBN 0-387-13610-X.

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