cyanic-selkie/wikianc huggingface dataset:
Emmy Noether
Göttingen
Noether 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. 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 tended to exclude 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 said of one such instance: "The enemy has been defeated; he has cleared out."
Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the building was closed for a state holiday, she gathered the class on the steps outside, led them through the woods, and lectured at a local coffee house. Later, after she had been dismissed by the Third Reich, she invited students into her home to discuss their plans for the future and mathematical concepts.
Moscow
thumb|205x205dp|[[Pavel Alexandrov]] In the winter of 1928–1929 Noether accepted an invitation to Moscow State University, where she continued working with P.S. Alexandrov. In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry. She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov, who later praised her contributions to the development of Galois theory.
hægri|thumb|Noether taught at the [[Moscow State University during the winter of 1928–1929.]] Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution. She was especially happy to see Soviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from a pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess".
Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933 he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry. Although this effort proved unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union. Meanwhile, her brother Fritz accepted a position at the Research Institute for Mathematics and Mechanics in Tomsk, in the Siberian Federal District of Russia, after losing his job in Germany, and was subsequently executed during the Great Purge.
Recognition
vinstri|thumb|Noether visited [[Zürich in 1932 to deliver a plenary address at the International Congress of Mathematicians.]] Noether's colleagues celebrated her fiftieth birthday in 1932, in typical mathematicians' 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. This pleased her immensely. He also sent her a mathematical riddle, which he called the "mμν-riddle of syllables". She solved immediately, but the riddle has been lost.
In November of the same year, Noether delivered a plenary address (großer Vortrag) on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich. The congress was attended by 800 people, including Noether's colleagues Hermann Weyl, Edmund Landau, and Wolfgang Krull. There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.
Expulsion from Göttingen by the Third Reich
When Adolf Hitler became the German Reichskanzler in January 1933, Nazi activity around the country increased dramatically. At the University of Göttingen the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by a privatdozent named Werner Weber, a former student of Noether. Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."
One of the first actions of Hitler's administration was the Law for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" by serving in World War I. In April 1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen." Several of Noether's colleagues, including Max Born and Richard Courant, also had their positions revoked.
Noether accepted the decision calmly, providing support for others during this difficult time. Hermann Weyl later wrote that "Emmy Noether—her courage, her frankness, her unconcern about her own fate, her conciliatory spirit—was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace." Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory. When one of her students appeared in the uniform of the Nazi paramilitary organization Sturmabteilung (SA), she showed no sign of agitation and, reportedly, even laughed about it later. This, however, was before the bloody events of Kristallnacht in 1938, and their praise from Propaganda Minister Joseph Goebbels.
Refuge at Bryn Mawr and Princeton, in America
hægri|thumb|[[Bryn Mawr College provided a welcoming home for Noether during the last two years of her life.]] As dozens of newly unemployed professors began searching for positions outside of Germany, their colleagues in the United States sought to provide assistance and job opportunities for them. Albert Einstein and Hermann Weyl were appointed by the Institute for Advanced Study in Princeton, while others worked to find a sponsor required for legal immigration. Noether was contacted by representatives of two educational institutions: Bryn Mawr College, in the United States, and Somerville College at the University of Oxford, in England. After a series of negotiations with the Rockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.
At Bryn Mawr, Noether met and befriended Anna Wheeler, who had studied at Göttingen just before Noether arrived there. Another source of support at the college was the Bryn Mawr president, Marion Edwards Park, who enthusiastically invited mathematicians in the area to "see Dr. Noether in action!" Noether and a small team of students worked quickly through van der Waerden's 1930 book Moderne Algebra I and parts of Erich Hecke's Theorie der algebraischen Zahlen (Theory of algebraic numbers).
In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen. She also worked with and supervised Abraham Albert and Harry Vandiver. However, she remarked about Princeton University that she was not welcome at "the men's university, where nothing female is admitted".
Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects. In the summer of 1934 she briefly returned to Germany to see Emil Artin and her brother Fritz before he left for Tomsk. Although many of her former colleagues had been forced out of the universities, she was able to use the library as a "foreign scholar".
Death
hægri|thumb|250x250dp|Noether's remains were placed under the walkway surrounding the cloisters of Bryn Mawr's [[Bryn Mawr College#M. Carey Thomas Library|M. Carey Thomas Library.]] In April 1935 doctors discovered a tumor in Noether's pelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst "the size of a large cantaloupe". Two smaller tumors in her uterus appeared to be benign and were not removed, to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse on the fourth. On 14 April she fell unconscious, her temperature soared to 109 °F, 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."
A few days after Noether's death her friends and associates at Bryn Mawr held a small memorial service at College President Park's house. Hermann Weyl and Richard Brauer traveled from Princeton and spoke with Wheeler and Taussky about their departed colleague. In the months that followed, written tributes began to appear around the globe: Albert Einstein joined van der Waerden, Weyl, and Pavel Alexandrov in paying their respects. Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.
Contributions to mathematics and physics
Noether's work in abstract algebra and topology was influential in mathematics, while in physics, Noether's theorem has consequences for theoretical physics and dynamical systems. She showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways. Her friend and colleague Hermann Weyl described her scholarly output in three epochs: "Emmy Noether's scientific production fell into three clearly distinct epochs:
(1) the period of relative dependence, 1907–1919
(2) the investigations grouped around the general theory of ideals 1920–1926
(3) the study of the non-commutative algebras, their representations by linear transformations, and their application to the study of commutative number fields and their arithmetics"
In the first epoch (1907–1919), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. After moving to Göttingen in 1915, she produced her work for physics, the two Noether's theorems.
In the second epoch (1920–1926), Noether devoted herself to developing the theory of mathematical rings.
In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields.
Although the results of Noether's first epoch were impressive and useful, her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B.L. van der Waerden in their obituaries of her.
In these epochs, she was not merely applying ideas and methods of earlier mathematicians; rather, she was crafting new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a completely new theory of ideals in rings, generalizing earlier work of Richard Dedekind. She is also renowned for developing ascending chain conditions, a simple finiteness condition that yielded powerful results in her hands. Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as elimination theory and the algebraic varieties that had been studied by her father.
Historical context
In the century from 1832 to Noether's death in 1935, the field of mathematics – specifically algebra – underwent a profound revolution, whose reverberations are still being felt. Mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g., cubic, quartic, and quintic equations, as well as on the related problem of constructing regular polygons using compass and straightedge. Beginning with Carl Friedrich Gauss's 1832 proof that prime numbers such as five can be factored in Gaussian integers, Évariste Galois's introduction of permutation groups in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), William Rowan Hamilton's discovery of quaternions in 1843, and Arthur Cayley's more modern definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by ever-more-universal rules. Noether's most important contributions to mathematics were to the development of this new field, abstract algebra.
Background on abstract algebra and begriffliche Mathematik (conceptual mathematics)
A group consists of a set of elements and a single operation which combines a first and a second element and returns a third. The operation must satisfy certain constraints for it to determine a group: It must be closed (when applied to any pair of elements of the associated set, the generated element must also be a member of that set), it must be associative, there must be an identity element (an element which, when combined with another element using the operation, results in the original element, such as adding zero to a number or multiplying it by one), and for every element there must be an inverse element.
A ring likewise, has a set of elements, but now has two operations. The first operation must make the set a group, and the second operation is associative and distributive with respect to the first operation. It may or may not be commutative; this means that the result of applying the operation to a first and a second element is the same as to the second and first – the order of the elements does not matter. If every non-zero element has a multiplicative inverse (an element x such that a x = x a = 1 ), the ring is called a division ring. A field is defined as a commutative division ring.
Groups are frequently studied through group representations. In their most general form, these consist of a choice of group, a set, and an action of the group on the set, that is, an operation which takes an element of the group and an element of the set and returns an element of the set. Most often, the set is a vector space, and the group represents symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. This is a type of symmetry of space, because space itself does not change when it is rotated even though the positions of objects in it do. Noether used these sorts of symmetries in her work on invariants in physics.
A powerful way of studying rings is through their modules. A module consists of a choice of ring, another set, usually distinct from the underlying set of the ring and called the underlying set of the module, an operation on pairs of elements of the underlying set of the module, and an operation which takes an element of the ring and an element of the module and returns an element of the module.
The underlying set of the module and its operation must form a group. A module is a ring-theoretic version of a group representation: Ignoring the second ring operation and the operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring itself. An important special case of this is an algebra. (The word algebra means both a subject within mathematics as well as an object studied in the subject of algebra.) An algebra consists of a choice of two rings and an operation which takes an element from each ring and returns an element of the second ring. This operation makes the second ring into a module over the first. Often the first ring is a field.
Words such as "element" and "combining operation" are very general, and can be applied to many real-world and abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For example, the elements might be computer data words, where the first combining operation is exclusive or and the second is logical conjunction. Theorems of abstract algebra are powerful because they are general; they govern many systems. It might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay Noether's gift to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions. In his obituary of Noether, her student van der Waerden recalled that "The maxim by which Emmy Noether was guided throughout her work might be formulated as follows: "Any relationships between numbers, functions, and operations become transparent, generally applicable, and fully productive only after they have been isolated from their particular objects and been formulated as universally valid concepts.""
This is the begriffliche Mathematik (purely conceptual mathematics) that was characteristic of Noether. This style of mathematics was consequently adopted by other mathematicians, especially in the (then new) field of abstract algebra.
Example: Integers as a ring
The integers form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication. Any pair of integers can be added or multiplied, always resulting in another integer, and the first operation, addition, is commutative, i.e., for any elements a and b in the ring, a + b = b + a. The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that a combined with b might be different from b combined with a. Examples of noncommutative rings include matrices and quaternions. The integers do not form a division ring, because the second operation cannot always be inverted; there is no integer a such that 3 × a = 1.
The integers have additional properties which do not generalize to all commutative rings. An important example is the fundamental theorem of arithmetic, which says that every positive integer can be factored uniquely into prime numbers. Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem, now called the Lasker–Noether theorem, for the ideals of many rings. Much of Noether's work lay in determining what properties do hold for all rings, in devising novel analogs of the old integer theorems, and in determining the minimal set of assumptions required to yield certain properties of rings.
First epoch (1908–1919): Algebraic invariant theory
The discriminant is called "invariant" because it is not changed by linear substitutions x → a x + b y, y → c x + d y with determinant a d − b c = 1. These substitutions form the special linear group SL2.
One can ask for all polynomials in A, B, and C that are unchanged by the action of SL2; these are called the invariants of binary quadratic forms and turn out to be the polynomials in the discriminant.
More generally, one can ask for the invariants of homogeneous polynomials A0 xr y0 + ... + Ar x0 yr of higher degree, which will be certain polynomials in the coefficients A0, ..., Ar, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.
One of the main goals of invariant theory was to solve the "finite basis problem". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called generators, and then, adding or multiplying the generators together. For example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.
Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables. He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables. Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the special orthogonal group.
First epoch (1908–1919): Galois theory
Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable x of degree n, in which the coefficients are drawn from some ground field, which might be, for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not be choices of x, which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. If the polynomial is x2 + 1 and the field is the real numbers, then the polynomial has no roots, because any choice of x makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.
Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, +i and −i, where i is the imaginary unit, that is, i2 = −1. More generally, the extension field in which a polynomial can be factored into its roots is known as the splitting field of the polynomial.
The Galois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial. (In mathematical jargon, these transformations are called automorphisms.) The Galois group of x2 + 1 consists of two elements: The identity transformation, which sends every complex number to itself, and complex conjugation, which sends +i to −i. Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a permutation of the n roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups of the Galois group.
In 1918, Noether published a paper on the inverse Galois problem. Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "Noether's problem", which asks whether the fixed field of a subgroup G of the permutation group Sn acting on the field k(x1, ..., xn) always is a pure transcendental extension of the field k. (She first mentioned this problem in a 1913 paper, where she attributed the problem to her colleague Fischer.) She showed this was true for n = 2, 3, or 4. In 1969, R.G. Swan found a counter-example to Noether's problem, with n = 47 and G a cyclic group of order 47 (although this group can be realized as a Galois group over the rationals in other ways). The inverse Galois problem remains unsolved.
First epoch (1908–1919): Physics
For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the system is responsible for the conservation law. As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.
Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon:"If the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments."
Second epoch (1920–1926): Ascending and descending chain conditions
In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions. A sequence of non-empty subsets A1, A2, A3, etc. of a set S is usually said to be ascending, if each is a subset of the next
Conversely, a sequence of subsets of S is called descending if each contains the next subset:
A chain becomes constant after a finite number of steps if there is an n such that A_n = A_m for all m ≥ n. A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.
Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects—and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage.
For example: How to use chain conditions to show that every set of sub-objects has a maximal/minimal element or that a complex object can be generated by a smaller number of elements. These conclusions often are crucial steps in a proof.
Pascal (forritunarmál)
Pascal er stefjumál sem var þróað af Niklaus Wirth árið 1970 sem forritunarmál sem myndi henta vel í mótaða forritun. Pascal á sér ættingja Object Pascal sem er sniðið að hlutbundinni forritun.
Emmy Noether
Second epoch (1920–1926): Ascending and descending chain conditions
Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called Noetherian in her honor. By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. A Noetherian module is a module in which every strictly ascending chain of submodules becomes constant after a finite number of steps. A Noetherian space is a topological space in which every strictly ascending chain of open subspaces becomes constant after a finite number of steps; this definition makes the spectrum of a Noetherian ring a Noetherian topological space.
Pascal (forritunarmál)
Saga
Pascal er byggt á forritunarmálinu ALGOL og er nefnt eftir og til heiðurs stærðfræðingnum og heimsspekingnum Blaise Pascal. Forritunarmálin Modula-2 og Oberon voru þróuð út frá Pascal og eru svipuð.
Emmy Noether
Second epoch (1920–1926): Ascending and descending chain conditions
The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and, mutatis mutandis, the same holds for submodules and quotient modules of a Noetherian module. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. The chain condition also may be inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.
Pascal (forritunarmál)
Saga
Í upphafi var Pascal þróað sem forritunarmál ætlað til að kenna nemendum mótaða forritun. Margar kynslóðir nemenda hafa “slitið barnsskónum” í forritun með Pascal sem fyrsta mál í grunnáföngum forritunar til margra ára. Pascal er til í hinum ýmsu útfærslum í dag, til dæmis Free Pascal sem er meðal annars til bæði í 32-bita og 64-bita útgáfum. Allar gerðir Pascal henta til bæði kennslu og hugbúnaðarþróunar.
Emmy Noether
Second epoch (1920–1926): Ascending and descending chain conditions
Another application of such chain conditions is in Noetherian induction—also known as well-founded induction—which is a generalization of mathematical induction. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose that S is a partially ordered set. One way of proving a statement about the objects of S is to assume the existence of a counterexample and deduce a contradiction, thereby proving the contrapositive of the original statement. The basic premise of Noetherian induction is that every non-empty subset of S contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the minimal counterexample. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.
Pascal (forritunarmál)
Saga
Hlutar af fyrsta Macintosh stýrikerfinu voru handþýddir yfir í Motorola 68000 smalamál úr Pascal kóða sem var skrifaður fyrir Lisa stýrikerfið, forvera Macintosh (Seinni útgáfur notuðust einnig við töluvert af C++ kóða). Fyrstu ár Macintosh var Pascal aðal æðra-stigs forritunarmálið í hugbúnaðargerð fyrir Makkann.
Útfærslur
Fyrsti þýðandinn fyrir Pascal var hannaður í Zurich fyrir CDC 6000 series tölvukerfin og varð nothæfur 1970.
Emmy Noether
Second epoch (1920–1926): Commutative rings, ideals, and modules
Noether's paper, Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921), is the foundation of general commutative ring theory, and gives one of the first general definitions of a commutative ring. Before her paper, most results in commutative algebra were restricted to special examples of commutative rings, such as polynomial rings over fields or rings of algebraic integers. Noether proved that in a ring which satisfies the ascending chain condition on ideals, every ideal is finitely generated. In 1943, French mathematician Claude Chevalley coined the term, Noetherian ring, to describe this property. A major result in Noether's 1921 paper is the Lasker–Noether theorem, which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings. The Lasker–Noether theorem can be viewed as a generalization of the fundamental theorem of arithmetic which states that any positive integer can be expressed as a product of prime numbers, and that this decomposition is unique.
Noether's work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields, 1927) characterized the rings in which the ideals have unique factorization into prime ideals as the Dedekind domains: integral domains that are Noetherian, 0- or 1-dimensional, and integrally closed in their quotient fields. This paper also contains what now are called the isomorphism theorems, which describe some fundamental natural isomorphisms, and some other basic results on Noetherian and Artinian modules.
Pascal (forritunarmál)
Útfærslur
Fyrsti bandaríski Pascal þýðandinn var settur saman í Háskólanum í Illinois undir stjórn Donald B. Gillies fyrir PDP-11 og skilaði út hreinu vélamáli.
Í Zurich var búinn til “port pakki” til þess að hraða útbreiðslu tungumálsins. Í pakkanum var þýðandi sem bjó til kóða fyrir sýndarvél auk túlks fyrir “p-kóða” kerfið. Þó svo að “p-kóðinn” hafi í upphafi verið hannaður til að vera þýddur yfir í hreint smalamál þá er í það minnsta eitt kerfi, UCSD útfærslan, sem nýtti það í að búa til stefjumálskerfið UCSD p-System. P-kerfis þýðendur eru merktir P1-P4, þar sem P1 er elsta útgáfan og P4 sú nýjasta.
Emmy Noether
Second epoch (1920–1926): Elimination theory
In 1923–1924, Noether applied her ideal theory to elimination theory in a formulation that she attributed to her student, Kurt Hentzelt. She showed that fundamental theorems about the factorization of polynomials could be carried over directly. Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, usually by the method of resultants.
Pascal (forritunarmál)
Útfærslur
Watcom Pascal var þróað fyrir IBM stórtölvur í upphafi níunda áratugarins.
Emmy Noether
Second epoch (1920–1926): Elimination theory
For illustration, a system of equations often can be written in the form  M v = 0   where a matrix (or linear transform)   M   (without the variable x) times a vector v (that only has non-zero powers of x) is equal to the zero vector, 0. Hence, the determinant of the matrix  M   must be zero, providing a new equation in which the variable x has been eliminated.
Pascal (forritunarmál)
Útfærslur
IP Pascal var útfærsla af Pascal forritunarmálinu sem notaðist við Micropolis DOS, en var fljótlega færst yfir í CP/M sem keyrði á Z80.
Emmy Noether
Second epoch (1920–1926): Invariant theory of finite groups
Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper, Noether found a solution to the finite basis problem for a finite group of transformations  G   acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is called Noether's bound. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field is coprime to  |G|! (the factorial of the order  |G|   of the group G). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number  |G|, but Noether was not able to determine whether this bound was correct when the characteristic of the field divides   |G|! but not  |G|. For many years, determining the truth or falsehood of this bound for this particular case was an open problem, called "Noether's gap". It was finally solved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.
Pascal (forritunarmál)
Útfærslur
Apple bjó til sýna eigin útfærslu, Lisa Pascal fyrir Lisa námsstefnuna árið 1982 og 1985 færði Apple þennan þýðanda yfir á Macintosh og MPW. Sama ár skilgreindi Larry Tesler í samráði við Niklaus Wirth Object Pascal og voru þeim viðbótum bætt inn í Lisa Pascal og Mac Pascal þýðendurna.
Emmy Noether
Second epoch (1920–1926): Invariant theory of finite groups
In her 1926 paper, Noether extended Hilbert's theorem to representations of a finite group over any field; the new case that did not follow from Hilbert's work is when the characteristic of the field divides the order of the group. Noether's result was later extended by William Haboush to all reductive groups by his proof of the Mumford conjecture. In this paper Noether also introduced the Noether normalization lemma, showing that a finitely generated domain A over a field k has a set  { x1, ..., xn }   of algebraically independent elements such that A is integral over   k [x1, ... , xn].
Pascal (forritunarmál)
Útfærslur
Á níunda áratugnum skrifaði Anders Hejlsberg Blue Label Pascal þýðandann fyrir Nascom-2. Endurútfærsla af þessum þýðanda fyrir CP/M og IBM PC var markaðsett undir heitunum Compas pascal og PolyPascal áður en Borland eignaðist hann. Þýðandinn var endurskýrður Turbo Pascal og varð gríðarlega vinsæll, að hluta til vegna hagstæðs verðs og hins vegna þess að hann var einn fyrsti þýðandinn í þróunarumhverfi sem bauð upp á vinnusvæði sem náði yfir allan skjáinn. Anders Hejlsberg færði svo Turbo Pascal yfir á Macintosh 1986 og setti viðbæturnar úr Object Pascal yfir í Turbo Pascal. Þessar viðbætur skilu sér síðan inn í útgáfu 5.5 af Turbo Pascal fyrir PC.
Emmy Noether
Second epoch (1920–1926): Contributions to topology
Noether is credited with fundamental ideas that led to the development of algebraic topology from the earlier combinatorial topology, specifically, the idea of homology groups. According to the account of Alexandrov, Noether attended lectures given by Heinz Hopf and by him in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle" and he continues that, "When ... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly the groups of algebraic complexes and cycles of a given polyhedron and the subgroup of the cycle group consisting of cycles homologous to zero; instead of the usual definition of Betti numbers, she suggested immediately defining the Betti group as the complementary (quotient) group of the group of all cycles by the subgroup of cycles homologous to zero. This observation now seems self-evident. But in those years (1925–1928) this was a completely new point of view."
Pascal (forritunarmál)
Útfærslur
Borland ákvað hins vegar seinna að þeir vildu hlutbundnu eiginleikana aðeins fágaðri og hófu þróunina á Delphi umhverfinu og notuðu uppkastið af Obect Pascal frá Apple sem grunn. (Drögin frá Apple eru ekki opinber staðall). Borland kallaði þetta Obect Pascal í fyrstu útgáfunum af Delphi en breyttu nafninu formlega í Delphi í seinni útgáfum. Aðal viðbæturnar frá hlutbundnu eiginleikum eldra kerfisins voru tilvísanir á hluti, sýnadrsmiðir og eyðar, og eigindi. Margir aðrir þýðendur bjóða uppa á það sama, sjá Object Pascal.
Emmy Noether
Second epoch (1920–1926): Contributions to topology
Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others, and it became a frequent topic of discussion among the mathematicians of Göttingen. Noether observed that her idea of a Betti group makes the Euler–Poincaré formula simpler to understand, and Hopf's own work on this subject "bears the imprint of these remarks of Emmy Noether". Noether mentions her own topology ideas only as an aside in a 1926 publication, where she cites it as an application of group theory.
Pascal (forritunarmál)
Útfærslur
Super Pascal er útgáfa sem bætti við ótölulegum merkjum, “return” skipun og segðir sem tagnöfn.
Emmy Noether
Second epoch (1920–1926): Contributions to topology
hægri|thumb|[[Helmut Hasse worked with Noether and others to found the theory of central simple algebras.]]
Pascal (forritunarmál)
Halló heimur
Öll Pascal forrit byrja á "Program" lykilorðinu, valkvæmum lista of útværum skráalýsingum og síðan forritsbút merktum með " " og " " lykilorðunum. Semikommur skipta setningum, og punktur lætur forritið hætta. Pascal lítur sömu augum á há- og lágstafi.
Emmy Noether
Third epoch (1927–1935): Hypercomplex numbers and representation theory
Much work on hypercomplex numbers and group representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these results and gave the first general representation theory of groups and algebras.
Third epoch (1927–1935): Noncommutative algebra
Noether also was responsible for a number of other advances in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she founded the theory of central simple algebras.
A paper by Noether, Helmut Hasse, and Richard Brauer pertains to division algebras, which are algebraic systems in which division is possible. They proved two important theorems: a local-global theorem stating that if a finite-dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is trivial), and from this, deduced their Hauptsatz ("main theorem"):"every finite dimensional central division algebra over an algebraic number field F splits over a cyclic cyclotomic extension."These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra D are splitting fields. This paper also contains the Skolem–Noether theorem which states that any two embeddings of an extension of a field k into a finite-dimensional central simple algebra over k, are conjugate. The Brauer–Noether theorem gives a characterization of the splitting fields of a central division algebra over a field.
Assessment, recognition, and memorials
hægri|thumb|250x250dp|The Emmy Noether Campus at the [[University of Siegen is home to its mathematics and physics departments.]] Noether's work continues to be relevant for the development of theoretical physics and mathematics and she is consistently ranked as one of the greatest mathematicians of the twentieth century. In his obituary, fellow algebraist BL van der Waerden says that her mathematical originality was "absolute beyond comparison", and Hermann Weyl said that Noether "changed the face of algebra by her work". During her lifetime and even until today, Noether has been characterized as the greatest woman mathematician in recorded history by mathematicians such as Pavel Alexandrov, Hermann Weyl, and Jean Dieudonné.
In a letter to The New York Times, Albert Einstein wrote: "In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians."
On 2 January 1935, a few months before her death, mathematician Norbert Wiener wrote that "Miss Noether is ... 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."
At an exhibition at the 1964 World's Fair devoted to Modern Mathematicians, Noether was the only woman represented among the notable mathematicians of the modern world.
Noether has been honored in several memorials,
In fiction, Emmy Nutter, the physics professor in "The God Patent" by Ransom Stephens, is based on Emmy Noether.
Farther from home,
List of doctoral students
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Ragnar Bjarnason syngur með Svavari Gests og hljómsveit
Ragnar Bjarnason með Svavari Gests og hljómsveit er 45-snúninga hljómplata gefin út af Íslenzkum tónum árið 1961. Á henni flytja Ragnar Bjarnason og hljómsveit Svavars Gests tvö lög. Platan er hljóðrituð í mono. Upptaka: Ríkisútvarpið. Umslag: Amatörverslunin ljósmyndastofa. Pressun: AS Nera í Osló.
Pascal (forritunarmál)
Halló heimur
Hér er dæmi um kóða í mjög einföldu forriti: program HelloWorld(output);
begin writeln('Halló heimur!') end.
Gagnaskipan
Í Pascal eru einföldu tögin real, integer, character og boolean auk enumeration, sem var kynnt sem nýtt tag í Pascal.
program myprog; var r: real; i: integer; c: char; b: boolean; e: (one, two, three, four, five);
Hægt er að búa til undirsvið úr raðtögum.
var x: 1..10; y: 'a'..'z'; z: two..four;
Hægt er að búa til tög úr öðrum tögum með tagskilgreiningu.
program myotherprog; type x = integer; y = x; ...
Fleiri flóknari tög er hægt að setja saman úr einfaldari tögum:
type a = array [1..10] of integer; b = record a: integer; b: char end; c = file of a;
Bendlar
Pascal styður notkun benda: type a = ^b; b = record a: integer; b: char; c: a      end;