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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. 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] |
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] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' refers only to ''univariate equations'', that is polynomial equations that involve only one variable (mathematics), variable. On the other hand, a polynomial equation may involve several variables. In the case of several variables (the ''multivariate'' case), the term ''polynomial equation'' is usually preferred to ''algebraic equation''. For example, :x^5-3x+1=0 is an algebraic equation with integer coefficients and :y^4 + \frac - \frac + xy^2 + y^2 + \frac = 0 is a multivariate polynomial equation over the rationals. Some but not all polynomial equations with Rational number, rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formally Real Field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. Alternative definitions The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as (infinitely many) first-order sentences in the language of fields and are equivalent to the above definition. A formally real field ''F'' is a field that also satisfies one of the following equivalent properties:Milnor and Husemoller (1973) p.60 * −1 is not a sum of squares in ''F''. In other words, the Stufe of ''F'' is infinite. (In particular, such a field must have characteristic 0, since in a field of characteristic ''p'' the element −1 is a sum of 1s.) This can be expressed in first-order logic by \forall x_1 (-1 \ne x_1^2), \forall x_1 x_2 (-1 \ne x_1^2 + x_2^2), etc., with one sentence for each number of va ... [...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 . 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 closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chevalley–Warning Theorem
In number theory, the Chevalley–Warning theorem implies that certain polynomial, polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Emil Artin, Artin's and Leonard Eugene Dickson, Dickson's conjecture that finite fields are quasi-algebraically closed fields . Statement of the theorems Let \mathbb be a finite field and \_^r\subseteq\mathbb[X_1,\ldots,X_n] be a set of polynomials such that the number of variables satisfies :n>\sum_^r d_j where d_j is the Degree of a polynomial#Extension to polynomials with two or more variables, total degree of f_j. The theorems are statements about the solutions of the following system of polynomial equations :f_j(x_1,\dots,x_n)=0\quad\text\, j=1,\ldots, r. * The ''Chevalley–Warning theorem'' states that the number of common solutions (a_1,\dots,a_n) \in \mathbb^n is di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ''L'' over ''K''. A subset ''S'' of ''L'' is a transcendence basis of ''L'' / ''K'' if it is algebraically independent over ''K'' and if furthermore ''L'' is an algebraic extension of the field ''K''(''S'') (the field obtained by adjoining the elements of ''S'' to ''K''). One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension and is denoted trdeg''K'' ''L'' or trdeg(''L'' / ''K''). If no field ''K'' is specified, the transcendence degree of a field ''L'' is its degree relative to the prime field of the same characteristic, i.e., the rational numbers field Q if ''L'' is of characteristic 0 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm Form
In mathematics, a norm form is a homogeneous form in ''n'' variables constructed from the field norm of a field extension ''L''/''K'' of degree ''n''. That is, writing ''N'' for the norm mapping to ''K'', and selecting a basis ''e''1, ..., ''e''''n'' for ''L'' as a vector space over ''K'', the form is given by :''N''(''x''1''e''1 + ... + ''x''''n''''e''''n'') in variables ''x''1, ..., ''x''''n''. In number theory norm forms are studied as Diophantine equations, where they generalize, for example, the Pell equation.. For this application the field ''K'' is usually the rational number field, the field ''L'' is an algebraic number field, and the basis is taken of some order in the ring of integers ''O''''L'' of ''L''. See also *Trace form In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''. Definition Let ''K'' be a field and ''L'' a finite extension (and hence a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
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 groups of an algebraic curve. The theorem was published by Chiungtze C. Tsen Chiungtze C. Tsen (; Chang-Du Gan: sɛn˦˨ tɕjuŋ˨˩˧ tsɹ̩˦˨ April 2, 1898 – October 1, 1940), given name Chiung (), was a Chinese mathematician born in Nanchang, Jiangxi. He is known for his work in algebra. He was one of Emmy No ... in 1933. See also * Tsen rank References * * * * Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
