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Tsen's Theorem
In mathematics, Tsen's theorem states that a function field ''K'' of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups ''H'' ''i''(''K'', ''K''*) vanish for ''i'' ≥ 1. This result is used to calculate the étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ... groups of an algebraic curve. The theorem was published by Chiungtze C. Tsen in 1933. See also * Tsen rank References * * * * Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. Every field K is contained in an algebraically closed field C, and the roots in C of the polynomials with coefficients in K form an algebraically closed field called an algebraic closure of K. Given two algebraic closures of K there are isomorphisms between them that fix the elements of K. Algebraically closed fields appear in the following chain of class inclusions: Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x^2+1=0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-algebraically Closed
In mathematics, a 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 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 .... 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'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Galois Cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a natural way on some abelian groups, for example those constructed directly from ''L'', but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type. History Étale cohomology was introduced by , using some suggestions by Jean-Pierre Serre, and was motivated by the attempt to construct a Weil cohomology theory in order to prove the Weil conjectures. The foundations were soon after worked out by Grothendieck together with Michael Artin, and published as and SGA 4. Grothendieck used étale cohomology to prove some of the Weil conjectures (Bernard Dwork had already managed to prove the rationality par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsen Rank
In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their study in 1936. We consider a system of ''m'' polynomial equations in ''n'' variables over a field ''F''. Assume that the equations all have constant term zero, so that (0, 0, ... ,0) is a common solution. We say that ''F'' is a T''i''-field if every such system, of degrees ''d''1, ..., ''d''''m'' has a common non-zero solution whenever :n > d_1^i + \cdots + d_m^i. \, The ''Tsen rank'' of ''F'' is the smallest ''i'' such that ''F'' is a T''i''-field. We say that the Tsen rank of ''F'' is infinite if it is not a T''i''-field for any ''i'' (for example, if it is formally real). Properties * A field has Tsen rank zero if and only if it is algebraically closed. * A finite field has Tsen rank 1: this is the Chevalley–Warning theorem. * If ''F'' is algebrai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rocky Mountain Journal Of Mathematics
'' The Rocky Mountain Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Rocky Mountain Mathematics Consortium. Founded in 1971, the journal publishes both research and expository articles on mathematics, with an emphasis on survey articles. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.25. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 0.250. References External links * Mathematics journals Academic journals established in 1971 English-language journals Bimonthly journals Academic journals published by learned and professional societies {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
