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Philipp Furtwängler
Friederich Pius Philipp Furtwängler (April 21, 1869 – May 19, 1940) was a German number theorist. Biography Furtwängler wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms (''Zur Theorie der in Linearfaktoren zerlegbaren ganzzahligen ternären kubischen Formen''), under Felix Klein. Most of his academic life, from 1912 to 1938, was spent at the University of Vienna, where he taught for example Kurt Gödel, who later said that Furtwängler's lectures on number theory were the best mathematical lectures that he ever heard; Gödel had originally intended to become a physicist but turned to mathematics partly as a result of Furtwängler's lectures. Furtwängler was paralysed and, without notes, lectured from a wheelchair while his assistant wrote equations on the blackboard. Some of Furtwängler's doctoral students were Wolfgang Gröbner, Nikolaus Hofreiter, Henry Mann, Otto Schreier, and Olga Taussky-Todd. Through these and others, he has over 30 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal Theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called Principalization (algebra), ''principalization'', or sometimes ''capitulation''. Formal statement For any algebraic number field ''K'' and any ideal (ring theory), ideal ''I'' of the ring of integers of ''K'', if ''L'' is the Hilbert class field of ''K'', then :IO_L\ is a principal ideal α''O''''L'', for ''O''''L'' the ring of integers of ''L'' and some element α in it. History The principal ideal theorem was conjectured by , and was the last remaining aspect of his program on class fields to be completed, in 1929. reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1869 Births
Events January–March * January 3 – Abdur Rahman Khan is defeated at Tinah Khan, and exiled from Afghanistan. * January 5 – Scotland's oldest professional football team, Kilmarnock F.C., is founded. * January 20 – Elizabeth Cady Stanton is the first woman to testify before the United States Congress. * January 21 – The P.E.O. Sisterhood, a philanthropic educational organization for women, is founded at Iowa Wesleyan College in Mount Pleasant, Iowa. * January 27 – The Republic of Ezo is proclaimed on the northern Japanese island of Ezo (which will be renamed Hokkaidō on September 20) by remaining adherents to the Tokugawa shogunate. * February 5 – Prospectors in Moliagul, Victoria, Australia, discover the largest alluvial gold nugget ever found, known as the "Welcome Stranger". * February 20 – Ranavalona II, the Merina Queen of Madagascar, is baptized. * February 25 – The Iron and Steel Institute is formed in Lon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Austrian Mathematicians
Austrian may refer to: * Austrians, someone from Austria or of Austrian descent ** Someone who is considered an Austrian citizen, see Austrian nationality law * Austrian German dialect * Something associated with the country Austria, for example: ** Austria-Hungary ** Austrian Airlines (AUA) ** Austrian cuisine ** Austrian Empire ** Austrian monarchy ** Austrian German (language/dialects) ** Austrian literature ** Austrian nationality law ** Austrian Service Abroad ** Music of Austria ** Austrian School of Economics * Economists of the Austrian school of economic thought * The Austrian Attack variation of the Pirc Defence chess opening. See also * * * Austria (other) * Australian (other) * L'Autrichienne (other) is the feminine form of the French word , meaning "The Austrian". It may refer to: *A derogatory nickname for Queen Marie Antoinette of France *L'Autrichienne (film), ''L'Autrichienne'' (film), a 1990 French film on Marie Antoinette wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MacTutor History Of Mathematics Archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathematicians, as well as information on famous curves and various topics in the history of mathematics. The History of Mathematics archive was an outgrowth of Mathematical MacTutor system, a HyperCard database by the same authors, which won them the European Academic Software award in 1994. In the same year, they founded their web site. it has biographies on over 2800 mathematicians and scientists. In 2015, O'Connor and Robertson won the Hirst Prize of the London Mathematical Society for their work... The citation for the Hirst Prize calls the archive "the most widely used and influential web-based resource in history of mathematics". See also * Mathematics Genealogy Project * MathWorld * PlanetMath PlanetMath is a free, collaborative, m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principalization (algebra)
In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 afte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer–Vandiver Conjecture
In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime ''p'' does not divide the class number ''hK'' of the maximal real subfield K=\mathbb(\zeta_p)^+ of the ''p''-th cyclotomic field. The conjecture was first made by Ernst Kummer on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, reprinted in , and independently rediscovered around 1920 by Philipp Furtwängler and , As of 2011, there is no particularly strong evidence either for or against the conjecture and it is unclear whether it is true or false, though it is likely that counterexamples are very rare. Background The class number ''h'' of the cyclotomic field \mathbb(\zeta_p) is a product of two integers ''h''1 and ''h''2, called the first and second factors of the class number, where ''h''2 is the class number of the maximal real subfield K=\mathbb(\zeta_p)^+ of the ''p''-th cyclotomic field In number theory, a cyclotomic field is a number field obtaine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keller's Conjecture
In geometry, Keller's conjecture is the conjecture that in any tiling of -dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire -dimensional face with each other. For instance, in any tiling of the plane by identical squares, some two squares must share an entire edge, as they do in the illustration. This conjecture was introduced by , after whom it is named. A breakthrough by showed that it is false in ten or more dimensions, and after subsequent refinements, it is now known to be true in spaces of dimension at most seven and false in all higher dimensions. The proofs of these results use a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. The related Minkowski lattice cube-tiling conjecture states that whenever a tiling of space by identical cubes has the additional property that the cubes' centers form a lattice, some cubes must meet face-to-face. It was proved by György Haj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Class Field
In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonically isomorphic to the ideal class group of ''K'' using Frobenius elements for prime ideals in ''K''. In this context, the Hilbert class field of ''K'' is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every real embedding of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E''). Examples *If the ring of integers of ''K'' is a unique factorization domain, in particular if K = \mathbb , then ''K'' is its own Hilbert class field. *Let K = \mathbb(\sqrt) of discriminant -15. The field L = \mathbb(\sqrt, \sqrt) has discriminant 225=(-15)^2 and so is an everywhere unramified extension of ''K'', and it is abelian. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Reciprocity
In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by , though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839. Background and notation Let m > 1 be an integer, and let \mathcal_m be the ring of integers of the ''m''-th cyclotomic field \mathbb(\zeta_m), where \zeta_m=e^ is a primitive ''m''-th root of unity. The numbers \zeta_m, \zeta_m^2,\dots\zeta_m^m=1 are units in \mathcal_m. (There are other units as well.) Primary numbers A number \alpha\in\mathcal_m is called primary if it is not a unit, is relatively prime to m, and is congruent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and diophantine geometry (Hasse principle), and to local zeta functions. Life Hasse was born in Kassel, Province of Hesse-Nassau, the son of Judge Paul Reinhard Hasse, also written Haße (12 April 1868 – 1 June 1940, son of Friedrich Ernst Hasse and his wife Anna Von Reinhard) and his wife Margarethe Louise Adolphine Quentin (born 5 July 1872 in Milwaukee, daughter of retail toy merchant Adolph Quentin (b. May 1832, probably Berlin, Kingdom of Prussia) and Margarethe Wehr (b. about 1840, Prussia), then raised in Kassel). After serving in the Imperial German Navy in World War I, he studied at the University of Göttingen, and then at the University of Marburg under Kurt Hensel, writing a dissertation in 1921 containing the Hasse–Minkowsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
