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Quasi-algebraically Closed Field
In mathematics, a field (mathematics), field ''F'' is called quasi-algebraically closed (or ''C''1) 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 hypersurface defined by ''P'', in projective space of degree , then has a point over ''F''. Examples * ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Brauer Group
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer. The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles. Construction A central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' such that ''A'' is a simple ring and the center of ''A'' is equal to ''K''. Note that CSAs are in general ''not'' division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be Irreducible component, irreducible, which means that it is not the Union (set theory), union of two smaller Set (mathematics), sets that are Closed set, closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Zariski Topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in 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] [Amazon] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), 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 definable set, 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 Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Guy Terjanian
Guy Terjanian is a French mathematician who has worked on algebraic number theory. He achieved his Ph.D. under Claude Chevalley in 1970, and at that time published a counterexample to the original form of a conjecture of Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ..., 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 References Further readingmath.unicaen.frarticle ''Topic: Arithmetic & geometry'' French people of Armenian descent 20th-century French ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P-adic Field
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 it is a definite quadratic form. More explicitly, 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 a definite 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. Over the real numbers, more generally in the case where ''F'' is a real closed field (so that the signature is defined), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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'' that 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 e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Division Ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element usually denoted , such that . So, (right) ''division'' may be defined as , but this notation is avoided, as one may have . A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field). All division rings are simple. That is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
