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Quasi-algebraically Closed 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 b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Group
Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Brauer (1929–2021), Austrian painter, poet, and actor, father of Timna Brauer * August Brauer (1863-1917), German zoologist * Friedrich Moritz Brauer (1832–1904), Austrian entomologist and museum director * Georg Brauer (1908–2001), German chemist * Ingrid Arndt-Brauer (born 1961), German politician; member of the Bundestag * Jono Brauer (born 1981), Australian Olympic skier * Max Brauer (1887–1973), German politician; First Mayor of Hamburg * Michael Brauer (contemporary), American audio engineer * Rich Brauer (born 1954), American politician from Illinois; state legislator since 2003 * Richard Brauer (1901–1977), German-American mathematician * Richard H. W. Brauer (contemporary), American art museum director; eponym of the Bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Extension
In field theory, a branch of algebra, a primary extension ''L'' of ''K'' is a field extension such that the algebraic closure of ''K'' in ''L'' is purely inseparable over ''K''.Fried & Jarden (2008) p.44 Properties * An extension ''L''/''K'' is primary if and only if it is linearly disjoint from the separable closure of ''K'' over ''K''. * A subextension of a primary extension is primary. * A primary extension of a primary extension is primary (transitivity). * Any extension of a separably closed field is primary. * An extension is regular if and only if it is separable and primary. * A primary extension of a perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is ... is regular. References * Field (mathematics) {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subvariety
A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms. Plant taxonomy Subvariety is ranked: *below that of variety (''varietas'') *above that of form (''forma''). Subvariety is an infraspecific taxon. Name Its name consists of three parts: #a genus name (genera) #a specific epithet (species) #an infraspecific epithet (subvariety) To indicate the subvariety rank, the abbreviation "subvar." is put before the infraspecific epithet. References Species In biology, a species is the basic unit of classification and a taxonomic rank of an organism, as well as a unit of biodiversity. A species is often defined as the largest group of organisms in which any two individuals of the appropriate s ... Plant taxonomy Biology terminology {{Botany-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
P-adic Field
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Quadratic Form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector space ''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that is quadratic space and ''W'' is a subspace of ''V''. Then ''W'' is called an isotropic subspace of ''V'' if ''some'' vector in it is isotropic, a totally isotropic subspace if ''all'' vectors in it are isotropic, and an anisotropic subspace if it does not contain ''any'' (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. A quadratic form ''q'' on a finite-dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Norm
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the Weyl algebra K ,\partial_X/math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.) For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or ''Brauer equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
