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. Events 1174 - Jocelin, Abbot of Melrose, is elected Bishop of Glasgow. Year 1882 ( MDCCCLXXXII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common Events 43 BC - Battle of Forum Gallorum: Mark Antony, besieging Julius Caesar 's assassin Decimus Junius Brutus in Year 1935 ( MCMXXXV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. PLEASE TAKE NOTE************ A mathematician is a person whose primary area of study and research is the field of Mathematics. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Often described as the most important woman in the history of mathematics,[1][2] she revolutionized the theories of rings, fields, and algebras. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with 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. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves [3]
Born in the Bavarian town of Erlangen to the noted mathematician Max Noether, Emmy originally planned to teach French and English after passing the required examinations. Bavaria ( German:, with an area of 70553 Km² (27241 square miles and almost 12 Erlangen is a Middle Franconian City in Bavaria, Germany. It is located at the confluence of the river Regnitz and its large tributary Max Noether ( 24 September 1844 - 13 December 1921) was a German Mathematician who worked on Algebraic geometry and She did not pursue languages, however, and studied mathematics at the University of Erlangen, where her father lectured. History The university was founded in 1742 in Bayreuth by Frederick Margrave of Bayreuth, and moved to Erlangen in 1743 After completing her dissertation in 1907 under the supervision of Paul Albert Gordan, she worked at the Mathematical Institute without pay for seven years. Paul Albert Gordan ( 27 April 1837 &ndash 21 December 1912) was a German Mathematician, a student of Carl Jacobi
In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group The University of Göttingen ( German: Georg-August-Universität Göttingen) is a University in the city of Göttingen, Germany. 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. Habilitation is the highest academic qualification a person can achieve by their own pursuit in certain European and Asian countries Private docent (abbreviates PD or Priv-Doz) is a title conferred in some European university systems especially in German -speaking countries 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". Göttingen ( ˈgœtɪŋən, Low German: Chöttingen is a College town in Lower Saxony, Germany. 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. Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, 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 International Congress of Mathematicians (ICM is the largest congress in the Mathematics community Zürich (, Zürich German: Züri, Zurich, Zurigo; in English generally Zurich) is the largest city in Switzerland and capital of the 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. Bryn Mawr College ( brin-mar is a highly selective women's liberal arts college located in Bryn Mawr, a community in Lower Merion The Commonwealth of Pennsylvania ( often colloquially referred to as PA (its abbreviation by natives and Northeasterners is a state located in the Northeastern In 1935, she underwent surgery for an ovarian cyst and, despite signs of speedy recovery, died four days later at the age of 53. An ovarian cyst is any collection of fluid surrounded by a very thin wall within an Ovary.
Noether's mathematical work has been divided into three "epochs". In the first (1908–19), she made valuable contributions to the theories of algebraic invariants and number fields. Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Her seminal work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has [4] However, the second epoch (1920–26) marks the beginning of her groundbreaking work that "changed the face of algebra". Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules [5] In her classic paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921), Noether developed the theory of ideals in commutative rings into a powerful tool with wide-ranging applications. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In this epoch, Noether elegantly used the ascending chain condition, and objects satisfying it are named Noetherian in her honor. The ascending chain condition (ACC and descending chain condition (DCC are finiteness properties satisfied by certain algebraic structures most importantly ideals In the third epoch (1927–35), she published major works on noncommutative algebras, as well as united hypercomplex numbers and the representation theory of groups with the theory of modules and ideals. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In addition to her own publications, Noether was generous with her ideas, and she is credited with several novel lines of research published by other mathematicians, even in disparate fields such as algebraic topology. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic
Contents |
Biography
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. Poliomyelitis, often called polio or infantile paralysis, is an acute viral Infectious disease spread from person to person primarily via Largely self-taught, he received a doctorate from the University of Heidelberg in 1868. Autodidacticism (also autodidactism) is self-education or self-directed learning A doctorate is an Academic degree that indicates the highest level of academic achievement The Ruprecht Karl University of Heidelberg ( University of Heidelberg, Ruperto Carola, Heidelberg University, or simply Heidelberg) is a 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. Erlangen is a Middle Franconian City in Bavaria, Germany. It is located at the confluence of the river Regnitz and its large tributary [6] As a mathematician, Max Noether contributed mainly to algebraic geometry, following in the footsteps of Alfred Clebsch. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Rudolf Friedrich Alfred Clebsch ( 19 January 1833 – 7 November 1872) was a German Mathematician who made important contributions 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. In Mathematics, in the theory of Algebraic curves certain divisors on a curve C are particular in the sense of determining more compatible functions than In Algebraic geometry, a field of Mathematics, the AF+BG theorem is a result of Max Noether which describes when the equation of an Algebraic curve
Emmy Noether was born on 23 March 1882, the first of four children. Events 1174 - Jocelin, Abbot of Melrose, is elected Bishop of Glasgow. Year 1882 ( MDCCCLXXXII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common 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. Myopia (from Greek: μυωπία myopia "near-sightedness" also called near- or short-sightedness, is a refractive defect A lisp ( OE wlisp, stammering is a Speech impediment, historically also known as sigmatism. 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. [7] 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. [8]
Of her three brothers, only Fritz Noether, born in 1884, is remembered for his academic accomplishments. Fritz Alexander Ernst Noether (October 7 1884 in Erlangen – September 10 1941 in Orel, Russia) was a German Mathematician After studying in Munich, he made a reputation for himself in the field of applied mathematics. Munich (München; Minga is the capital city of Bavaria, Germany. Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains 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. [9]
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. History The university was founded in 1742 in Bayreuth by Frederick Margrave of Bayreuth, and moved to Erlangen in 1743 This was an unconventional decision; two years earlier, the Academic Senate of the university had declared that allowing coeducation would "overthrow all academic order". Mixed-sex education, (or just Mixed education) also known as Coeducation, is the integrated education to males and females at the same school facilities [10] 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. An audit is an educational term in the United States for the completion of a course of study for which no assessment is made or grade awarded Despite the obstacles, on 14 July 1903 she passed the graduation exam at a Realgymnasium in Nuremberg. Events 1223 - Louis VIII becomes King of France upon the death of his father Philip II of France. Year 1903 ( MCMIII) was a Common year starting on Thursday (link will display calendar of the Gregorian calendar or a Common year starting The Gymnasium is the classical higher or secondary schools of Germany for gifted students [11]
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. Karl Schwarzschild ( October 9, 1873 - May 11, 1916) was a German Jewish Physicist and Astronomer. Hermann Minkowski ( June 22 1864 – January 12 1909) was a Russian born German Mathematician, of Jewish Ludwig Otto Blumenthal ( July 20, 1876 – November 12, 1944) was a German Mathematician and professor at RWTH Aachen Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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. Events 69 - Second Battle of Bedriacum, forces under Antonius Primus the commander of the Danube armies loyal to Vespasian, defeat Year 1904 ( MCMIV) was a Leap year starting on Friday (link will display calendar of the Gregorian calendar (or a Leap year starting on Working under the supervision of Paul Albert Gordan, she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms, 1907). Paul Albert Gordan ( 27 April 1837 &ndash 21 December 1912) was a German Mathematician, a student of Carl Jacobi Although it was well received, Noether later referred to her thesis as "crap" and "a jungle of formulas". [12]
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. Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect She also worked with Erhard Schmidt and Ernst Fischer, sometimes discussing advanced concepts with Fischer by mailing commentary written on postcards. Erhard Schmidt ( January 13, 1876 – December 6, 1959) was a German Mathematician born in Dorpat (now Ernst Sigismund Fischer ( July 12, 1875 - November 14, 1954) was born in Vienna Austria. [13]
University of Göttingen
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. See Comparative linguistics for the narrower field of "comparative philology" See also History An historian is an individual who studies and writes about History, and is regarded as an Authority on it Private docent (abbreviates PD or Priv-Doz) is a title conferred in some European university systems especially in German -speaking countries 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?"[14] 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. "[14]
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. The German Army (Deutsches Heer heɐ) is the land component of the armed forces of the Federal Republic of Germany. World War I (abbreviated WWI; also known as the First World War, the Great War, and the War to End All She returned to Erlangen for several weeks, mostly to care for her aging father. [15]
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. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these [16] 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". Leon Max Lederman (born July 15, 1922) is an American Experimental physicist and Nobel Prize in Physics laureate for Christopher T Hill (born 1951 is a theoretical physicist at the Fermi National Accelerator Laboratory. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry [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 (an untenured professor with limited internal administrative rights and functions[17]). 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. [18]
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. In Mathematics, ideal theory is the theory of ideals in Commutative rings and is the precursor name for the contemporary subject of Commutative In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. The following year she published a landmark paper called Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regard to ideals. The ascending chain condition (ACC and descending chain condition (DCC are finiteness properties satisfied by certain algebraic structures most importantly ideals Canadian mathematician Irving Kaplansky has called this work "revolutionary";[19] it gave rise to the term "Noetherian ring". Irving Kaplansky ( March 22, 1917 &ndash June 25, 2006) was a Canadian Mathematician. In Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals. [20]
Soon afterwards, she began supervising doctoral students, including Grete Hermann, who later spoke reverently of her "dissertation-mother". Grete Hermann (1901-1984 was a German Mathematician and Philosopher. [21] 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. Max Deuring ( 9 December 1907, Göttingen, Germany – 20 December 1984, Göttingen Germany was a mathematician This is a glossary of arithmetic and Diophantine geometry in Mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts Hans Fitting ( 13 November 1906 München-Gladbach (now Mönchengladbach) – 15 June 1938 Königsberg (now Kaliningradwas Fitting's theorem is a mathematical Theorem proved by Hans Fitting. The Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in Abstract algebra. Zeng Jiongzhi ( April 2 1898 – 1940) also known as Chiungtze C In mathematics Tsen's theorem states that a function field K of an Algebraic curve over an algebraically closed field is Quasi-algebraically closed. She also worked closely with Wolfgang Krull, originator of Krull's theorem. In Mathematics, more specifically in Ring theory, Krull's theorem, named after Wolfgang Krull, proves the existence of Maximal ideals in any [22]
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. [23] 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". [24]
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. Gottfried Emanuel Noether ( Karlsruhe, Germany 1915 - August 22 1991, Willimantic, Connecticut) was an American [25] 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". Olga Taussky Todd ( August 30, 1906, Olomouc, then Austria-Hungary - October 7, 1995, Pasadena, California [26] 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. [27]
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!"[28] 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. "[29] 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. [30] Later, after she had been dismissed by the Third Reich, she invited students into her home to discuss their future plans and mathematical concepts. Nazi Germany and the Third Reich are the common English names for Germany under the regime of Adolf Hitler and the National Socialist German Workers [31]
In 1924, the young Dutch mathematician Bartel Leendert van der Waerden arrived at the University of Göttingen. Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, He began working immediately with Noether, who provided invaluable methods of abstract conceptualization. He said later that her originality was "absolute beyond comparison". [32] 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. Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician Noether". [33] She sometimes allowed her colleagues and students to receive credit for her ideas, helping them develop their careers at the expense of her own. [34]
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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Pavel Sergeyevich Alexandrov (Па́вел Серге́евич Алекса́ндров sometimes romanized Aleksandroff or Aleksandrov ( November 16 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. The Rockefeller Foundation (RF is a prominent Philanthropic organization and Private foundation based at 420 Fifth Avenue New York City. [35] 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". [36]
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, she taught classes in abstract algebra and algebraic geometry. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory. Lev Semenovich Pontryagin ( Russian Лев Семёнович Понтрягин ( 3 September 1908 &ndash 3 May 1988) was a Nikolai Chebotaryov (often spelled Chebotarov or Chebotarev) (Николай Григорьевич Чеботарёв Микола Григорович Чоботарьов In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory [37]
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. See also Russian Revolution (1905 The Russian Revolution of 1916 refers to a series of popular revolutions in Russia, and the events surrounding them 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. A soviet (сове́т, "council" originally was a workers' local council in late Imperial Russia. The Bolsheviks, originally also Bolshevists ( Большевик Большевист (singular, derived from bolshe, "more" were a faction 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". A pension is a family-owned Guesthouse or Boarding house. This term is used in Spain, Italy and other countries as a synonym of cheap Hostel [38]
Following 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. Moscow (Москва́ romanised: Moskvá, IPA: see also other names) is the Capital and the largest city of Narkompros (Наркомпрос is an abbreviation for the People's Commissariat for Education (Народный комиссариат просвещения the Although this effort was unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union. [38] Her brother Fritz, meanwhile, took a position at the Research Institute for Mathematics and Mechanics in Tomsk, Siberia after losing his job in Germany. Fritz Alexander Ernst Noether (October 7 1884 in Erlangen – September 10 1941 in Orel, Russia) was a German Mathematician Tomsk (Томск is a city on the Tom River in the southwest of Siberian Federal District, Russia, the administrative centre of Siberia (Сиби́рь Sibir) is the name given to the vast region constituting almost all of Northern Asia and for the most part currently serving [39]
Recognition
Noether and the Austrian mathematician Emil Artin were awarded the Ackermann–Teubner Memorial Award in 1932 for their contributions to mathematics. Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician 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. For a detailed discussion of the English translation of Reich, see Reich. 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[40] (full professor). [17]
Noether's 50th 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. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic The Mathematische Annalen (abbreviated as Math Ann or Math Annal In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability [41] 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. [40]
In September of the same year, Noether delivered a major address (großer Vortrag) at the International Congress of Mathematicians in Zürich, Switzerland. The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community Zürich (, Zürich German: Züri, Zurich, Zurigo; in English generally Zurich) is the largest city in Switzerland and capital of the Switzerland (English pronunciation; Schweiz Swiss German: Schwyz or Schwiiz Suisse Svizzera Svizra officially the Swiss Confederation The conference was attended by 800 people, with 420 officially participating. Notable participants included Hermann Weyl, Edmund Landau, and Wolfgang Krull. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Edmund Georg Hermann (Yehezkel Landau ( February 14, 1877 – February 19, 1938) was a German Jewish Mathematician 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. [42]
Expulsion
When Adolf Hitler became Chancellor of Germany in January 1933, Nazi activity around the country increased dramatically, including at the University of Göttingen. Hi and welcome to Wikipedia! Please understand that this article is frequently vandalized and vandalism is reverted immediately Nazism, which was a short name for National Socialism (Nationalsozialismus refers primarily to the Ideology and practices of the National Socialist German 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. Antisemitism (alternatively spelled anti-semitism or anti-Semitism; also rarely known as judeophobia) is the Prejudice against or hostility "[43] Several of Noether's colleagues, including Max Born and Richard Courant, had their positions revoked. Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum Richard Courant (born January 8, 1888 &ndash January 27, 1972) was a German American Mathematician. [44]
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. "[44] 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. "[43] As usual, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory. In Mathematics, class field theory is a major branch of Algebraic number 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. The, abbreviated SA, ( German for "Assault detachment" or "Assault section" usually translated as " stormtroop(ers [44]
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. Princeton University is a private Coeducational research university located in Princeton, New Jersey. Noether was contacted by representatives of two schools, Bryn Mawr College and the University of Oxford. Bryn Mawr College ( brin-mar is a highly selective women's liberal arts college located in Bryn Mawr, a community in Lower Merion The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the 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. The Rockefeller Foundation (RF is a prominent Philanthropic organization and Private foundation based at 420 Fifth Avenue New York City. [45]
At Bryn Mawr, Noether met and befriended Anna Johnson Pell Wheeler, who had studied at Göttingen just before Noether arrived there. Anna Johnson Pell Wheeler ( May 5, 1883 - 1966 was an American mathematician 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!"[46] 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). Erich Hecke ( September 20, 1887 &ndash February 13, 1947) was a German Mathematician. [47]
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". The Institute for Advanced Study, located in Princeton New Jersey, United States is a center for theoretical research [48] In addition to Abraham Flexner and Oswald Veblen, who had invited her, she worked with and supervised Abraham Adrian Albert and Harry Vandiver. Abraham Flexner ( November 13 1866, Louisville Kentucky - September 21 1959) was an American educator Oswald Veblen ( 24 June 1880 in Decorah Iowa &ndash 10 August, 1960) was an American Mathematician, Abraham Adrian Albert ( November 9, 1905 &ndash June 6, 1972) was a mathematician of Russian ancestry Harry Schultz Vandiver ( 21 October, 1882 &ndash 9 January, 1973) was an American Mathematician, known for work in [49] Her time in the United States was pleasant, surrounded as she was by supportive colleagues and ensconced in her favorite subjects. [50] 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". [51]
Death
In April 1935, doctors discovered a tumor in Noether's pelvis. See also Cancer A tumor or tumour is the name for a swelling or lesion formed by an abnormal growth of cells (termed neoplastic The pelvis (pl pelvises or pelves) or pelvic girdle is the irregular bony structure located at the base of the spine (properly known 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". An ovarian cyst is any collection of fluid surrounded by a very thin wall within an Ovary. Cantaloupe (also cantaloup) refers to two varieties of Muskmelon ( Cucumis melo), which is a Species in the family Cucurbitaceae (a [52] Two smaller tumors in her uterus appeared to be benign and were not removed, to avoid prolonging the surgery. The uterus (from the Latin word for womb) is the major Female reproductive organ of most Mammals including Humans One end the For three days she appeared to convalesce normally, and recovered quickly from a circulatory collapse on the fourth. This is an article about the rock music band "Circulatory System" On 14 April, she fell unconscious, her temperature soared to 109 °F (42. Events 43 BC - Battle of Forum Gallorum: Mark Antony, besieging Julius Caesar 's assassin Decimus Junius Brutus in 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. "[52]
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. Richard Dagobert Brauer ( February 10, 1901 &ndash April 17, 1977) was a leading German and American Mathematician 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. Bryn Mawr College ( brin-mar is a highly selective women's liberal arts college located in Bryn Mawr, a community in Lower Merion [53]
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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of She showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways. [54] Her friend and colleague Hermann Weyl described her scholarly output in three epochs. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. In the first epoch (1908–19), 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). Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect Paul Albert Gordan ( 27 April 1837 &ndash 21 December 1912) was a German Mathematician, a student of Carl Jacobi Mist is a phenomenon of small droplets suspended in Air. It can occur as part of natural Weather or Volcanic activity and is common in cold air above 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. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Ernst Sigismund Fischer ( July 12, 1875 - November 14, 1954) was born in Vienna Austria. After moving to Göttingen in 1915, she produced her seminal work for physics, the two Noether's theorems. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In the second epoch (1920–26), Noether devoted herself to developing the theory of mathematical rings. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real [55] In the third epoch (1927–35), Noether focused on noncommutative algebra, linear transformations, and commutative number fields. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that [56]
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. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. 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. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero In Mathematics, a quintic equation is a Polynomial Equation of degree five In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power General properties These properties apply to both convex and star regular polygons Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles 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. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 A Gaussian integer is a Complex number whose real and imaginary part are both Integers The Gaussian integers with ordinary addition and multiplication of complex In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Noether's most important contributions to mathematics were to the development of this new field, abstract algebra,[57] 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 CDs and DVDs, makes it possible to recover data even in the presence of small scratches on the disc. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Coding theory is one of the most important and direct applications of Information theory. Reed-Solomon error correction is an Error-correcting code that works by Oversampling a Polynomial constructed from the data A Compact Disc (also known as a CD) is an Optical disc used to store digital data, originally developed for storing digital audio DVD (also known as " Digital Versatile Disc " or " Digital Video Disc " - see Etymology)is
Despite the generality of abstract algebra, some elements may be understood by analogy to the integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Indeed, any pair of integers can be added or multiplied, always resulting in another integer, and much of Noether's work dealt with abstract systems, know as rings, that mimic the usual operations of addition and multiplication on the integers. Addition is the mathematical process of putting things together These operations must obey rules, some of which are very similar to the normal rules with integers. For example, the addition must be commutative (i. In Mathematics, commutativity is the ability to change the order of something without changing the end result e. , for any elements a and b in the ring, a + b = b + a). However, unlike the usual multiplication on integers, 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician 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. In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left 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. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. 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. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers This ring has 12 elements: 0, 1, 2, 3, and so on up to 11. The element 0 is usually written 12 on a clock, but they are the same in this ring as their difference, which is 12, is divisible by 12. The elements of this ring are the 12 residue classes associated with the ideal of 12 in the ring of integers. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for 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. In Computing, " word " is a term for the natural unit of data used by a particular computer design In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of 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. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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. In Mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection 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 (1908–19)
Algebraic invariant theory
Much of Noether's work in the first epoch was associated with invariant theory, principally algebraic invariant theory. Invariant theory is a branch of Abstract algebra that studies actions of groups on algebraic varieties from the point of view of their effect Invariant theory is concerned with expressions that remain constant (invariant) under a group of transformations. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. In Mathematics, the term "symmetric function" can mean two different things In Mathematics, a symmetric polynomial is a polynomial P ( X 1 X 2 &hellip X n) In several fields of Mathematics the term permutation is used with different but closely related meanings This article is about the zeros of a function which should not be confused with the value at zero. 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. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position g. , the cross-ratio of projective geometry. In Mathematics, the cross-ratio of a set of four distinct points on the Complex plane is given by (z_1z_2z_3z_4 = \frac{(z_1-z_3(z_2-z_4}{(z_1-z_4(z_2-z_3} Projective geometry is a non- metrical form of Geometry, notable for its principle of duality.
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. In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same 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 invariants of any set of homogenous rational functions over a field containing the rational numbers and under the action of the general linear group of invertible matrices. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation [58][59] However, 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 concerns transformations of number fields that permute the roots of an equation. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In several fields of Mathematics the term permutation is used with different but closely related meanings 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. When a Polynomial is expressed as a sum or difference of terms (e In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers There may or may not be choices of x which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. This article is about the zeros of a function which should not be confused with the value at zero. 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. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. 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. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial. In Abstract algebra, the splitting field of a Polynomial P ( X) over a given field K is a Field extension
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 Mathematics, a Galois group is a group associated with a certain type of Field extension. (In mathematical jargon, these transformations are called automorphisms. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself ) 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. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. 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. In several fields of Mathematics the term permutation is used with different but closely related meanings 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 Mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of Field extensions In its most basic In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of
In 1918, Noether published a seminal paper on the inverse Galois problem. In Mathematics, the inverse Galois problem concerns whether or not every Finite group appears as the Galois group of some Galois extension of [60] 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, . In Mathematics, a rational variety is an Algebraic variety, over a given field K, which is Birationally equivalent to Projective In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying . . , xn) is always a pure transcendental extension of the field k. In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i (She first mentioned this problem in a 1913 paper,[61] where she attributed the problem to her colleague Fischer. Ernst Sigismund Fischer ( July 12, 1875 - November 14, 1954) was born in Vienna Austria. ) 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[62] (though this group can be realized as a Galois group over the rationals in other ways). In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, a Galois group is a group associated with a certain type of Field extension. The inverse Galois problem is still unsolved.
Physics
Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In Mechanics, a constant of motion is a quantity that is conserved throughout the motion imposing in effect a constraint on the motion Göttingen ( ˈgœtɪŋən, Low German: Chöttingen is a College town in Lower Saxony, Germany. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Gravitation is a natural Phenomenon by which objects with Mass attract one another Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical 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. In Physics, the law of conservation of energy states that the total amount of Energy in an isolated system remains constant and cannot be created although it may 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. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has [63] 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. "[64]
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. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position [65] The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position 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. In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός
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. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world [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. [65]
Second epoch (1920–26)
Commutative rings, ideals, and modules
Her paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921)[66] is the foundation of general commutative ring theory, and gives one of the first general definitions of a commutative ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property [67] 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. The ascending chain condition (ACC and descending chain condition (DCC are finiteness properties satisfied by certain algebraic structures most importantly ideals In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In 1943, French mathematician Claude Chevalley coined the term Noetherian ring to describe this property. Claude Chevalley ( 11 February 1909, Johannesburg, South Africa - 28 June 1984, Paris) was a French In Abstract algebra, a Noetherian ring is a ring that satisfies the Ascending chain condition on ideals. [67] A Noetherian module is a module that satisfies the ascending chain condition on submodules, where the submodules are partially ordered by inclusion. In Abstract algebra, an Noetherian module is a module that satisfies the Ascending chain condition on its Submodules where the submodules are In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars (A Noetherian topological space is one that satisfies a similar ascending chain condition on open subsets, e. In Mathematics, a Noetherian topological space is a Topological space in which closed subsets satisfy the Descending chain condition. g. , the spectrum of a Noetherian ring; however, Noether never worked on these. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of ) 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. In Mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection 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. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1
Noether's work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields, 1927)[68] 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. In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper In Commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull ( 1899 - 1971) is defined to be the In Commutative algebra, the notions of an element integral over a ring (also called an algebraic integer over the ring and of an integral extension of 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. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Abstract algebra, an Artinian module is a module that satisfies the Descending chain condition on its submodules
Elimination theory
In 1923–24, Noether applied her ideal theory to elimination theory—in a formulation that she attributed to her student, Kurt Hentzelt[69]—showing that fundamental theorems about the factorization of polynomials could be carried over directly. In Commutative algebra and Algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between Polynomials of In Mathematics and Computer algebra, Polynomial Factorization typically refers to factoring a polynomial into Irreducible polynomials [70] Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, usually by the method of resultants. In Commutative algebra and Algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between Polynomials of In Mathematics, the resultant of two Monic polynomials P and Q over a field k is defined as the product 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. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n
Return to invariant theory
In her 1926 paper,[71] 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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 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 the Mumford conjecture. William Joseph Haboush is an American mathematician who is best known for his 1975 proof of one of David Mumford 's conjectures known as the Haboush's theorem There are several conjectures in mathematics by David Mumford. ) 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, . In Mathematics the Noether normalization lemma is a technical result of Commutative algebra, introduced in. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Abstract algebra, the transcendence degree of a Field extension L / K is a certain rather coarse measure of the "size" of the extension . . , xn such that A is integral over k[x1, . In Commutative algebra, the notions of an element integral over a ring (also called an algebraic integer over the ring and of an integral extension of . . , xn].
Third epoch (1927–35)
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. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Noether united the results and gave the first general representation theory of groups and algebras. [72] 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. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real This single work was of fundamental importance for the development of modern algebra. [73]
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. Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician Richard Dagobert Brauer ( February 10, 1901 &ndash April 17, 1977) was a leading German and American Mathematician Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic In Ring theory and related areas of Mathematics a central simple algebra ( CSA) over a field K (also called a Brauer algebra [74]
A seminal paper by Noether, Helmut Hasse and Richard Brauer pertains to division algebras,[75] which are algebraic systems in which division is possible. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic Richard Dagobert Brauer ( February 10, 1901 &ndash April 17, 1977) was a leading German and American Mathematician In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field 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. In Mathematics, Helmut Hasse 's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Ring theory and related areas of Mathematics a central simple algebra ( CSA) over a field K (also called a Brauer algebra In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. These theorems allow one to classify all finite dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra D are splitting fields. In Abstract algebra, the splitting field of a Polynomial P ( X) over a given field K is a Field extension [76] 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. In Mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in Ring theory which characterizes The Brauer–Noether theorem[77] gives a characterization of the splitting fields of a central division algebra over a field.
Assessment and memorials
Noether's work continues to be relevant for the development of theoretical physics and mathematics, and she is consistently ranked as one of the greatest mathematicians of the 20th century. In his obituary, fellow algebraist B. L. van der Waerden says that her mathematical originality was "absolute beyond comparison",[78] and Hermann Weyl said that Noether "changed the face of algebra by her work". Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules [5] Several respected mathematicians and physicists have characterized Noether as the greatest woman mathematician in recorded history. For example, in a letter to The New York Times, Albert Einstein wrote:[1]
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. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical 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.
On 2 January 1935, a few months before her death, mathematician Norbert Wiener wrote that[79]
Miss Noether is . Norbert Wiener ( November 26, 1894, Columbia Missouri – March 18, 1964, Stockholm, Sweden) was an American . . the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of Madame Curie.
In their obituaries, her colleagues Pavel Alexandrov[80] and Hermann Weyl[81] likewise stated that she was the greatest woman ever to work in the field. Pavel Sergeyevich Alexandrov (Па́вел Серге́евич Алекса́ндров sometimes romanized Aleksandroff or Aleksandrov ( November 16 Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. [2] This assessment has continued in the decades since her death. In a 1964 World's Fair exhibit entitled "Men of Modern Mathematics", Noether was the only female represented. Expo (short for "exposition" and also known as World Fair and World's Fair) is the name given to various large public exhibitions held since the History of the Mathematica Exhibition In March 1961 a new science wing at the California Museum of Science and Industry in Los Angeles opened [82] Modern biographical encyclopedias of mathematicians continue to characterize her as the greatest woman mathematician up to her time. [83] In 1983, mathematician Jean Dieudonné summarized the consensus that "she was by far the best woman mathematician of all time, and one of the greatest mathematicians (male or female) of the XXth century. Jean Alexandre Eugène Dieudonné ( July 1 1906, Lille - November 29 1992, Nice) was a French mathematician "[84]
Noether has been honored in several memorials. The Association for Women in Mathematics holds a Noether Lecture to honor women in mathematics every year; in its 2005 pamphlet for the event, the Association characterizes Noether as "one of the great mathematicians of her time, someone who worked and struggled for what she loved and believed in. The Association for Women in Mathematics (AWM is a non-profit organization devoted to promoting equal treatment and equal opportunity for women and girls in the mathematical sciences The Association for Women in Mathematics (AWM annually presents the Noether Lectures to honor women who have made fundamental and sustained contributions to the mathematical Her life and work remain a tremendous inspiration". [85] Consistent with her dedication to her students, the University of Siegen houses its mathematics and physics buildings on the Emmy Noether Campus. The University of Siegen (Universität Siegen is a German scientific University, located in Siegen, North Rhine-Westphalia. [86] A street in her hometown Erlangen has been named after her and her father, Max Noether; and the successor to the secondary school she attended in Erlangen has been renamed the Emmy Noether School. Max Noether ( 24 September 1844 - 13 December 1921) was a German Mathematician who worked on Algebraic geometry and [84] Further from home, the Nöther crater on the far side of the Moon is named for her, as is the 7001 Noether asteroid. Nöther is a lunar crater on the far side of the Moon. It is located in the far northern latitudes to the northwest of the Poczobutt Far Side of the Moon, in original French, La face cachée de la lune, is a 2003 film by Robert Lepage. 7001 Noether (1955 EH is a Main-belt Asteroid discovered on March 14, 1955 by Indiana University at Brooklyn. [87][88]
See also
Notes
- ^ a b Einstein, Albert. Emmy Noether is counted among the greatest mathematicians of all time "Professor Einstein Writes in Appreciation of a Fellow-Mathematician". 5 May 1935. Events 553 - The Second Council of Constantinople begins 1215 - Rebel Barons renounce their allegiance to King John Year 1935 ( MCMXXXV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. Online at the MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J Retrieved on 13 April 2008. Events 1111 - Henry V is crowned Holy Roman Emperor. 1204 - The Fourth Crusade sacks Constantinople 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common
- ^ a b Osen 1974, p. 152; Alexandrov 1981, p. 100.
- ^ a b Ne'eman, Yuval. Yuval Ne'eman (יובל נאמן born 14 May 1925, died 26 April 2006) was an Israeli Soldier, physicist "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. The American Mathematical Society (AMS is an association of professional Mathematicians dedicated to the interests of mathematical research and scholarship which OCLC 223099225. The OCLC Online Computer Library Center is according to its website a "nonprofit membership computer library service and research organization dedicated to the public purpose ISBN 978-0198510451. pp. 83–101.
- ^ a b Lederman & Hill 2004, p. 73.
- ^ a b Dicke 1981, p. 128
- ^ Kimberling 1981, pp. 3–5; Osen 1974, p. 142; Lederman & Hill 2004, pp. 70–71; Dick 1981, pp. 7–9.
- ^ Dick 1981, pp. 9–10.
- ^ Dick 1981, pp. 10–11; Osen 1974, p. 142.
- ^ Dick 1981, pp. 25, 45; Kimberling, p. 5.
- ^ Quoted in Kimberling 1981, p. 10.
- ^ Dick 1981, pp. 11–12; Kimberling 1981, pp. 8–10; Lederman & Hill 2004, p. 71.
- ^ 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.
- ^ Kimberling 1981, pp. 11–12; Dick 1981, pp. 18–24; Osen 1974, p. 143.
- ^ a b Kimberling 1981, p. 14; Dick 1981, p. 32; Osen 1974, pp. 144–145; Lederman & Hill 2004, p. 72.
- ^ Dick 1981, pp. 24–26.
- ^ Osen 1974, pp. 144–145; Lederman & Hill 2004, p. 72.
- ^ a b Dick 1981, p. 188.
- ^ Kimberling 1981, p. 14–18; Osen 1974, p. 145; Dick 1981, pp. 33–34.
- ^ Kimberling 1981, p. 18.
- ^ Kimberling 1981, p. 18; Dick 1981, pp. 44–45; Osen 1974, pp. 145–146.
- ^ Dick 1981, p. 51.
- ^ Dick 1981, pp. 53–57.
- ^ Dick 1981, pp. 37–49.
- ^ van der Waerden 1935, p. 98.
- ^ Dick 1981, pp. 46–48.
- ^ Taussky 1981, p. 80
- ^ Dick 1981, pp. 40–41.
- ^ Mac Lane 1981, p. 77; Dick 1981, p. 37.
- ^ Dick 1981, pp. 38–41.
- ^ Mac Lane 1981, p. 71
- ^ Dick 1981, p. 76
- ^ van der Waerden 1935, p. 100.
- ^ Dick 1981, pp. 57–58; Kimberling 1981, p. 19; Lederman & Hill 2004, p. 74.
- ^ Lederman & Hill 2004, p. 74; Osen 1974, p. 148.
- ^ Kimberling 1981, pp. 24–25; Dick 1981, pp. 61–63.
- ^ Alexandrov 1981, pp. 100, 107.
- ^ Dick 1981, pp. 63–64; Kimberling 1981, p. 26; Alexandrov 1981, pp. 108–110.
- ^ a b Alexandrov 1981, pp. 106–109.
- ^ Osen 1974, p. 150; Dick 1981, pp. 82–83.
- ^ a b Dick 1981, pp. 72–73; Kimberling 1981, pp. 26–27.
- ^ Hasse 1933
- ^ Kimberling 1981, p. 26–27; Dick 1981, pp. 74–75.
- ^ a b Kimberling 1981, p. 29.
- ^ a b c Dick 1981, pp. 75–76; Kimberling 1981, pp. 28–29.
- ^ Dick 1981, pp. 78–79; Kimberling 1981, pp. 30–31.
- ^ Kimberling 1981, pp. 32–33; Dick 1981, p. 80.
- ^ Dick 1981, pp. 80–81.
- ^ Dick 1981, p. 81.
- ^ Dick 1981, pp. 81–82.
- ^ Osen 1974, p. 151; Dick 1981, p. 83.
- ^ Dick 1981, p. 82; Kimberling 1981, p. 34.
- ^ a b Kimberling 1981, pp. 37–38.
- ^ Kimberling 1981, p. 39.
- ^ Osen 1974, pp. 148–149; Kimberling 1981, pp. 11–12.
- ^ Gilmer 1981, p. 131.
- ^ Kimberling 1981, pp. 10–23.
- ^ Noether GE (1987), Grinstein LS, Campbell PJ, ed. Gottfried Emanuel Noether ( Karlsruhe, Germany 1915 - August 22 1991, Willimantic, Connecticut) was an American , 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 - ^ Weyl 1944, pp. 618-621
- ^ Hilbert 1890, p. 531
- ^ Noether 1918.
- ^ Noether 1913.
- ^ Swan 1969.
- ^ Noether 1918b
- ^ Kimberling 1981, p. 13.
- ^ a b Lederman & Hill 2004, pp. 97–116.
- ^ Noether 1921.
- ^ a b Gilmer 1981, p. 133.
- ^ Noether 1927.
- ^ Noether, Emmy (1921), “Über eine Arbeit des im Kriege gefallenen K. Hentzelt zur Eliminationstheorie”, Jahresbericht der Deutschen Mathematiker-Vereinigung 30 (Abt. 2): 101, <http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=249024>
- ^ Noether, Emmy (1923), “Zur Theorie der Polynomideale und Resultanten”, Mathematische Annalen 88: 53–79, <http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=362882>
Noether, Emmy (1923), “Eliminationstheorie und allgemeine Idealtheorie”, Mathematische Annalen 90: 229–261, <http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=362964>
Noether, Emmy (1923), “Eliminationstheorie und Idealtheorie”, Jahresbericht der Deutschen Mathematiker-Vereinigung 33: 116–120, <http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=248880> - ^ Noether 1926.
- ^ Noether 1929.
- ^ van der Waerden 1985, p. 244.
- ^ Lam 1981, pp. 152–153.
- ^ Brauer, Hasse & Noether 1932.
- ^ Noether 1933.
- ^ Brauer & Noether 1927
- ^ Dicke 1981, p. 100
- ^ Kimberling 1981, p. 35
- ^ Dicke 1981, p. 154
- ^ Dicke 1981, p. 152
- ^ Duchin, Moon. "The Sexual Politics of Genius" (PDF). December 2004. University of Chicago. Retrieved on 13 April 2008. Events 1111 - Henry V is crowned Holy Roman Emperor. 1204 - The Fourth Crusade sacks Constantinople 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common
- ^ James I (2002), Remarkable Mathematicians from Euler to von Neumann, Cambridge: Cambridge University Press, pp. Ioan Mackenzie James (born May 23, 1928 Croydon, Surrey, England) is a British Mathematician working in 321, ISBN 0-521-81777-3
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. 1869, ISBN 0-684-19291-8 - ^ a b Noether GE (1987), Grinstein LS, Campbell PJ, ed. Gottfried Emanuel Noether ( Karlsruhe, Germany 1915 - August 22 1991, Willimantic, Connecticut) was an American , Women of Mathematics, New York: Greenwood press, p. p. 167, ISBN 0-313-24849-4
- ^ "Introduction". Profiles of Women in Mathematics: The Emmy Noether Lectures. Association for Women in Mathematics. The Association for Women in Mathematics (AWM is a non-profit organization devoted to promoting equal treatment and equal opportunity for women and girls in the mathematical sciences 2005. Retrieved on 13 April 2008. Events 1111 - Henry V is crowned Holy Roman Emperor. 1204 - The Fourth Crusade sacks Constantinople 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common
- ^ "Emmy-Noether-Campus". Universität Siegen. Retrieved on 13 April 2008. Events 1111 - Henry V is crowned Holy Roman Emperor. 1204 - The Fourth Crusade sacks Constantinople 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common
- ^ Schmadel 2003, p. 570.
- ^ Blue, Jennifer. Gazetteer of Planetary Nomenclature. USGS. The United States Geological Survey ( USGS) is a scientific agency of the United States government. 25 July 2007. Events 285 - Diocletian appoints Maximian as Caesar, co-ruler Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Retrieved on 13 April 2008. Events 1111 - Henry V is crowned Holy Roman Emperor. 1204 - The Fourth Crusade sacks Constantinople 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common
References
Selected works by Emmy Noether (in German)
- Noether, Emmy (1908), “Über die Bildung des Formensystems der ternären biquadratischen Form (On Complete Systems of Invariants for Ternary Biquadratic Forms)”, Journal für die reine und angewandte Mathematik 134: 23–90 and two tables, <http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261200>. Emmy Noether is counted among the greatest mathematicians of all time
- Noether, Emmy (1913), “Rationale Funkionenkörper (Rational Function Fields)”, J. Ber. d. DMV 22: 316–319, <http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=244058>.
- Noether, Emmy (1918), “Gleichungen mit vorgeschriebener Gruppe (Equations with Prescribed Group)”, Mathematische Annalen 78: 221–229, DOI 10. The Mathematische Annalen (abbreviated as Math Ann or Math Annal 1007/BF01457099.
- Noether, Emmy (1918b), “Invariante Variationsprobleme (Invariant Variation Problems)”, 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 (The Theory of Ideals in Ring Domains)”, Mathematische Annalen 83 (1), ISSN 0025-5831, <http://www.springerlink.com/content/m3457w8h62475473/fulltext.pdf>. An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication.
- Noether, Emmy (1926), “Der Endlichkeitsatz der Invarianten endlicher linearer Gruppen der Charakteristik p (Proof of the Finiteness of the Invariants of Finite Linear Groups of Characteristic p)”, Nachr. Ges. Wiss. 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 (Abstract Structure of the Theory of Ideals in Algebraic Number Fields)”, Mathematische Annalen 96 (1): 26–61, ISSN 0025-5831, <http://www.springerlink.com/content/v3t6331n8w244275/fulltext.pdf>. An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication.
- Brauer, Richard & Noether, Emmy (1927), “Über minimale Zerfällungskörper irreduzibler Darstellungen (On the Minimum Splitting Fields of Irreducible Representations)”, Sitz. Richard Dagobert Brauer ( February 10, 1901 &ndash April 17, 1977) was a leading German and American Mathematician Ber. d. Preuss. Akad. d. Wiss. : 221–228.
- Noether, Emmy (1929), “Hyperkomplexe Grössen und Darstellungstheorie (Hypercomplex Quantities and the Theory of Representations)”, Mathematische Annalen 30: 641–692, DOI 10. 1007/BF01187794.
- Brauer, Richard; Hasse, Helmut & Noether, Emmy (1932), “Beweis eines Hauptsatzes in der Theorie der Algebren (Proof of a Main Theorem in the Theory of Algebras)”, Journal für Math. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic 167: 399–404, <http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=260847>.
- Noether, Emmy (1933), “Nichtkommutative Algebren (Noncommutative Algebras)”, 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. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many
Additional sources
- Alexandrov, Pavel S. (1981), “In Memory of Emmy Noether”, in James W. Pavel Sergeyevich Alexandrov (Па́вел Серге́евич Алекса́ндров sometimes romanized Aleksandroff or Aleksandrov ( November 16 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>.
- Byers, Nina (2006), “Emmy Noether”, in Nina Byers and Gary Williams, Out of the Shadows: Contributions of 20th Century Women to Physics, Cambridge: Cambridge University Press, ISBN 0-5218-2197-5. Nina Byers is Research Professor and Professor of Physics Emeritus at UCLA.
- 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.
- Hasse, Helmut (1933), “Die Struktur der R. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic 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, Emmy Noether: A Tribute to Her Life and Work, New York: Marcel Dekker, Inc. , pp. 3–61, ISBN 0-8247-1550-0.
- Lam, Tsit Yuen (1981), “Representation Theory”, in James W. Brewer and Martha K. Smith, 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. Leon Max Lederman (born July 15, 1922) is an American Experimental physicist and Nobel Prize in Physics laureate for Christopher T Hill (born 1951 is a theoretical physicist at the Fermi National Accelerator Laboratory.
- Mac Lane, Saunders (1981), “Mathematics at the University of Göttingen 1831–1933”, in James W. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American 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.
- 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. Works Dictionary of minor planet names (5th Edition. Springer Verlag Berlin/Heidelberg 2003 ISBN 3-540-00238-3 Dictionary of minor planet ), Berlin: Springer-Verlag, ISBN 3-540-00238-3.
- Swan, Richard 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. Olga Taussky Todd ( August 30, 1906, Olomouc, then Austria-Hungary - October 7, 1995, Pasadena, California 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. Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich, 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.
- Weyl, Hermann (1944), “David Hilbert and his mathematical work”, Bulletin of the American Mathematical Society 50: 612–654, MR0011274, ISSN 0002-9904, DOI 10. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. Bulletin of the American Mathematical Society (often abbreviated as Bull Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. 1090/S0002-9904-1944-08178-0
- Hilbert, David (1890), “Über die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, ISSN 0025-5831, DOI 10. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most The Mathematische Annalen (abbreviated as Math Ann or Math Annal An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. 1007/BF01208503
External links
- "Invariante Variationsprobleme", Nachr. v. d. Ges. d. Wiss. zu Göttingen Original paper in German with link to English translation.
- "Emmy Noether" in CWP at UCLA
- Emmy Noether at the Mathematics Genealogy Project
- O'Connor, John J. The Mathematics Genealogy Project is a web-based Database that gives an Academic genealogy based on Dissertation supervision relations & Robertson, Edmund F. , “Emmy Noether”, MacTutor History of Mathematics archive
- Lebensläufe (German) Noether's application for admission to the University of Erlangen and three curricula vitae, two of which are shown in handwriting, with transcriptions. The MacTutor History of Mathematics archive is an award-winning website maintained by John J History The university was founded in 1742 in Bayreuth by Frederick Margrave of Bayreuth, and moved to Erlangen in 1743 The first of these is in Emmy Noether's own handwriting.
- Unpublished and published versions (German) of Noether's 1908 doctoral dissertation completed at Erlangen. A dissertation (also called thesis or disquisition) is a document that presents the author's Research and findings and is submitted in support of candidature
- Emmy Noether, Mentors & Colleagues (photo by Clark Kimberling)
- Oberwolfach collection of photos of Noether
- Correspondence between Noether and Helmut Hasse, 1925–35
| Persondata | |
|---|---|
| NAME | Noether, Emmy |
| ALTERNATIVE NAMES | Noether, Amalie; Emmy Noethe |
| SHORT DESCRIPTION | Mathematician |
| DATE OF BIRTH | 23 March 1882 |
| PLACE OF BIRTH | Erlangen, Germany |
| DATE OF DEATH | 14 April 1935 |
| PLACE OF DEATH | Bryn Mawr, Pennsylvania, United States |
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic