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Arason Invariant
In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial Discriminant#Discriminant of a quadratic form, discriminant and Clifford invariant over a Field (mathematics), field ''k'' of Characteristic (algebra), characteristic not 2, taking values in H3(''k'',Z/2Z). It was introduced by . The Rost invariant is a generalization of the Arason invariant to other algebraic groups. Definition Suppose that ''W''(''k'') is the Witt ring (forms), Witt ring of quadratic forms over a field ''k'' and ''I'' is the ideal of forms of even dimension. The Arason invariant is a group homomorphism from ''I''3 to the Galois cohomology group H3(''k'',Z/2Z). It is determined by the property that on the 8-dimensional diagonal form with entries 1, –''a'', –''b'', ''ab'', -''c'', ''ac'', ''bc'', -''abc'' (the 3-fold Pfister form«''a'',''b'',''c''») it is given by the cup product of the classes of ''a'', ''b'', ''c'' in H1(''k'',Z/2Z) = '' ... [...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] |
Cohomological Invariant
In mathematics, a cohomological invariant of an algebraic group ''G'' over a field is an invariant of forms of ''G'' taking values in a Galois cohomology group. Definition Suppose that ''G'' is an algebraic group defined over a field ''K'', and choose a separably closed field containing ''K''. For a finite extension ''L'' of ''K'' in let Γ''L'' be the absolute Galois group of ''L''. The first cohomology H1(''L'', ''G'') = H1(Γ''L'', ''G'') is a set classifying the ``G''-torsors over ''L'', and is a functor of ''L''. A cohomological invariant of ''G'' of dimension ''d'' taking values in a Γ''K''-module ''M'' is a natural transformation of functors (of ''L'') from H1(L, ''G'') to H''d''(L, ''M''). In other words a cohomological invariant associates an element of an abelian cohomology group to elements of a non-abelian cohomology set. More generally, if ''A'' is any functor from finitely generated extensions of a field to sets, then a cohomological invariant of ''A'' of dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial ax^2+bx+c is :b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Invariant
Clifford may refer to: People *Clifford (name), an English given name and surname, includes a list of people with that name *William Kingdon Clifford * Baron Clifford *Baron Clifford of Chudleigh * Baron de Clifford *Clifford baronets * Clifford family (bankers) * Jaryd Clifford * Justice Clifford (other) * Lord Clifford (other) Arts, entertainment, and media *'' Clifford the Big Red Dog'', a series of children's books **Clifford (character), the central character of ''Clifford the Big Red Dog'' ** ''Clifford the Big Red Dog'' (2000 TV series), 2000 animated TV series **'' Clifford's Puppy Days'', 2003 animated TV series **''Clifford's Really Big Movie'', 2004 animated movie ** ''Clifford the Big Red Dog'' (2019 TV series), 2019 animated TV series ** ''Clifford the Big Red Dog'' (film), 2021 live-action movie * ''Clifford'' (film), a 1994 film directed by Paul Flaherty * Clifford (Muppet) Mathematics *Clifford algebra, a type of associative algebra, named after W ... [...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] |
Characteristic (algebra)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rost Invariant
In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraic group ''G'' over a field ''k'', which associates an element of the Galois cohomology group H3(''k'', Q/Z(2)) to a principal homogeneous space for ''G''. Here the coefficient group Q/Z(2) is the tensor product of the group of roots of unity of an algebraic closure of ''k'' with itself. first introduced the invariant for groups of type F4 and later extended it to more general groups in unpublished work that was summarized by . The Rost invariant is a generalization of the Arason invariant. Definition Suppose that ''G'' is an absolutely almost simple simply connected algebraic group over a field ''k''. The Rost invariant associates an element ''a''(''P'') of the Galois cohomology group H3(''k'',Q/Z(2)) to a ''G''-torsor ''P''. The element ''a''(''P'') is constructed as follows. For any extension ''K'' of ''k'' there is an exact sequence :0\rightarrow H^3(K,\mathbf/\mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Ring (forms)
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field. Definition Fix a field ''k'' of characteristic not equal to two. All vector spaces will be assumed to be finite-dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.Milnor & Husemoller (1973) p. 14 Each class is represented by the core form of a Witt decomposition.Lorenz (2008) p. 30 The Witt group of ''k'' is the abelian group ''W''(''k'') of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms.Milnor & Husemoll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) where the group operation on the left side of the equation is that of ''G'' and on the right side that of ''H''. From this property, one can deduce that ''h'' maps the identity element ''eG'' of ''G'' to the identity element ''eH'' of ''H'', : h(e_G) = e_H and it also maps inverses to inverses in the sense that : h\left(u^\right) = h(u)^. \, Hence one can say that ''h'' "is compatible with the group structure". Older notations for the homomorphism ''h''(''x'') may be ''x''''h'' or ''x''''h'', though this may be confused as an index or a general subscript. In automata theory, sometimes homomorphisms are written to the right of their arguments without parentheses, so that ''h''(''x'') becomes simply xh. In areas of mathematics where one ... [...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 to 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 class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pfister Form
In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an ''n''-fold Pfister form over ''F'' is a quadratic form of dimension 2''n'' that can be written as a tensor product of quadratic forms :\langle\!\langle a_1, a_2, \ldots , a_n \rangle\!\rangle \cong \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle \otimes \cdots \otimes \langle 1, -a_n \rangle, for some nonzero elements ''a''1, ..., ''a''''n'' of ''F''. (Some authors omit the signs in this definition; the notation here simplifies the relation to Milnor K-theory, discussed below.) An ''n''-fold Pfister form can also be constructed inductively from an (''n''−1)-fold Pfister form ''q'' and a nonzero element ''a'' of ''F'', as q \oplus (-a)q. So the 1-fold and 2-fold Pfister forms look like: :\langle\!\langle a\rangle\!\rangle\cong \langle 1, -a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
