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38 relations: Algebraic function field, Algebraically closed field, Annals of Mathematics, Ax–Kochen theorem, Brauer group, Brauer's theorem on forms, Cambridge University Press, Central simple algebra, Characteristic (algebra), Chevalley–Warning theorem, Chiungtze C. Tsen, Cohomological dimension, Emil Artin, Emmy Noether, Field (mathematics), Finite field, Glossary of arithmetic and diophantine geometry, Graduate Texts in Mathematics, Guy Terjanian, Homogeneous polynomial, Hypersurface, Isotropic quadratic form, Local Fields, Mathematics, Model theory, P-adic number, Perfect field, Primary extension, Princeton University, Projective space, Pseudo algebraically closed field, Serge Lang, Springer Science+Business Media, Subvariety, Transcendence degree, Tsen rank, Tsen's theorem, Zariski topology.
Algebraic function field
In mathematics, an (algebraic) function field of n variables over the field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K.
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Algebraically closed field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.
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Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.
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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 d2+1 variables has a nontrivial zero.
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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.
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Brauer's theorem on forms
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.
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K. In other words, any simple algebra is a central simple algebra over its center.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Chevalley–Warning theorem
In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions.
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Chiungtze C. Tsen
Chiungtze C. Tsen (April 2, 1898 – October 1, 1940) was a Chinese mathematician born in Nanchang, Jiangxi, who proved Tsen's theorem.
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Cohomological dimension
In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations.
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Emil Artin
Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
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Emmy Noether
Amalie Emmy NoetherEmmy is the Rufname, the second of two official given names, intended for daily use.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
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Glossary of arithmetic and diophantine geometry
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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Guy Terjanian
Guy Terjanian is a French mathematician who has worked on algebraic number theory.
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Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.
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Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.
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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.
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Local Fields
Corps Locaux by Jean-Pierre Serre, originally published in 1962 and translated into English as Local Fields by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields, ramification, group cohomology, and local class field theory.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Model theory
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.
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P-adic number
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.
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Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds.
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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.
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Princeton University
Princeton University is a private Ivy League research university in Princeton, New Jersey.
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Projective space
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.
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Pseudo algebraically closed field
In mathematics, a field K is pseudo algebraically closed if it satisfies certain properties which hold for any algebraically closed field.
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Serge Lang
Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Subvariety
A subvariety (Latin: subvarietas) in botanical nomenclature is a taxonomic rank.
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Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension.
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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.
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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).
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Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.
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Redirects here:
C1 field, C2 field, Quasi algebraically closed, Quasi algebraically closed field, Quasi-algebraic closure, Quasi-algebraically closed.
References
[1] https://en.wikipedia.org/wiki/Quasi-algebraically_closed_field
