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Ax–Kochen Theorem
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer ''d'' there is a finite set ''Yd'' of prime numbers, such that if ''p'' is any prime not in ''Yd'' then every homogeneous polynomial of degree ''d'' over the p-adic numbers in at least ''d''2 + 1 variables has a nontrivial zero. The proof of the theorem The proof of the theorem makes extensive use of methods from mathematical logic, such as model theory. One first proves Serge Lang's theorem, stating that the analogous theorem is true for the field F''p''((''t'')) of formal Laurent series over a finite field F''p'' with Y_d = \varnothing. In other words, every homogeneous polynomial of degree ''d'' with more than ''d''2 variables has a non-trivial zero (so F''p''((''t'')) is a C2 field). Then one shows that if two Henselian valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic 0, then they are elementarily equi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Ax
James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made groundbreaking contributions in algebra and number theory using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in Number Theory, which was awarded for a series of three joint papers on Diophantine problems. Education and career James Ax graduated from Peter Stuyvesant High School in New York City and then the Brooklyn Polytechnic University. He earned his Ph.D. from the University of California, Berkeley in 1961 under the direction of Gerhard Hochschild, with a dissertation on ''The Intersection of Norm Groups''. After a year at Stanford University, he joined the mathematics faculty at Cornell University. He spent the academic year 1965–1966 at Harvard University on a Guggenheim Fellowship. In 1969, he moved from Cornell to the mathematics department at Stony Brook University and remained on the faculty until 1977, when he retired from his aca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Arithmetica
''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences The Institute of Mathematics of the Polish Academy of Sciences is a research institute of the Polish Academy of Sciences.Online archives (Library of Science, Issues: 1935–2000) 1935 establishments in Poland [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit marg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-algebraic Closure
In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . The idea itself is attributed to Lang's advisor Emil Artin. Formally, if ''P'' is a non-constant homogeneous polynomial in variables :''X''1, ..., ''X''''N'', and of degree ''d'' satisfying :''d'' < ''N'' then it has a non-trivial zero over ''F''; that is, for some ''x''''i'' in ''F'', not all 0, we have :''P''(''x''''1'', ..., ''x''''N'') = 0. In geometric language, the defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer's Theorem On Forms
:''There also is Brauer's theorem on induced characters.'' In mathematics, Brauer's theorem, named for Richard Brauer, is a result on the representability of 0 by forms over certain fields in sufficiently many variables. Statement of Brauer's theorem Let ''K'' be a field such that for every integer ''r'' > 0 there exists an integer ψ(''r'') such that for ''n'' ≥ ψ(r) every equation :(*)\qquad a_1x_1^r+\cdots+a_nx_n^r=0,\quad a_i\in K,\quad i=1,\ldots,n has a non-trivial (i.e. not all ''x''''i'' are equal to 0) solution in ''K''. Then, given homogeneous polynomials ''f''1,...,''f''''k'' of degrees ''r''1,...,''r''''k'' respectively with coefficients in ''K'', for every set of positive integers ''r''1,...,''r''''k'' and every non-negative integer ''l'', there exists a number ω(''r''1,...,''r''''k'',''l'') such that for ''n'' ≥ ω(''r''1,...,''r''''k'',''l'') there exists an ''l''-dimensional affine subspace ''M'' of ''Kn'' (regarded as a vector space over ''K'') satisfying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guy Terjanian
Guy Terjanian is a French mathematician who has worked on algebraic number theory. He achieved his Ph.D. under Claude Chevalley in 1966, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitably modified had just been proved as the Ax-Kochen theorem. In 1977, he proved that if ''p'' is an odd prime number, and the natural numbers ''x'', ''y'' and ''z'' satisfy x^ + y^ = z^, then ''2p'' must divide ''x'' or ''y''.G. Terjanian, ''Sur l'equation x^+ y^ = z^ ','' CR. Acad. Sc. Paris. ,. 285. (1977), 973-975. See also *Ax–Kochen theorem The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer ''d'' there is a finite set ''Yd'' of prime numbers, such that if ''p'' is any prime not in ''Yd'' then every homogeneous polynomial of degree '' ... References Further readingmath.unicaen.frarticle ''Topic: Arithmetic & geometry'' French people of Armenian descent 20th-century French m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C2 Field
In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . The idea itself is attributed to Lang's advisor Emil Artin. Formally, if ''P'' is a non-constant homogeneous polynomial in variables :''X''1, ..., ''X''''N'', and of degree ''d'' satisfying :''d'' < ''N'' then it has a non-trivial zero over ''F''; that is, for some ''x''''i'' in ''F'', not all 0, we have :''P''(''x''''1'', ..., ''x''''N'') = 0. In geometric language, the defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma had ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Louis Colliot-Thélène
Jean-Louis Colliot-Thélène (born 2 December 1947), is a French mathematician. He is a Directeur de Recherches at CNRS at the Université Paris-Saclay in Orsay. He studies mainly number theory and arithmetic geometry. Awards *Prize of the French Academy of Sciences " Charles Louis de Saulces de Freycine" (1985) *Invited Speaker to the International Congress of Mathematicians (Berkeley 1986) *Fermat Prize for mathematical research (1991) *Grand prize of the French Academy of Sciences " Léonid Frank" (2009) *Fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ... (2012) [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Denef
Jan Denef (born 4 September 1951) is a Belgian Belgian may refer to: * Something of, or related to, Belgium * Belgians, people from Belgium or of Belgian descent * Languages of Belgium, languages spoken in Belgium, such as Dutch, French, and German *Ancient Belgian language, an extinct languag ... mathematician. He is an Emeritus Professor of Mathematics at the KU Leuven, Katholieke Universiteit Leuven (KU Leuven). Denef obtained his PhD from KU Leuven in 1975 with a thesis on Hilbert's tenth problem; his advisors were Louis Philippe Bouckaert and Willem Kuijk. He is a specialist of model theory, number theory and algebraic geometry. He is well known for his early work on Hilbert's tenth problem and for developing the theory of motivic integration in a series of papers with François Loeser. He has also worked on computational number theory. Recently he proved a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem. In 2002 Denef was an List of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
