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Linked Field
In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property. Linked quaternion algebras Let ''F'' be a field of characteristic not equal to 2. Let ''A'' = (''a''1,''a''2) and ''B'' = (''b''1,''b''2) be quaternion algebras over ''F''. The algebras ''A'' and ''B'' are linked quaternion algebras over ''F'' if there is ''x'' in ''F'' such that ''A'' is equivalent to (''x'',''y'') and ''B'' is equivalent to (''x'',''z''). The Albert form for ''A'', ''B'' is :q = \left\langle\right\rangle \ . It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''. The quaternion algebras are linked if and only if the Albert form is isotropic. Linked fields The field ''F'' is ''linked'' if any two quaternion algebras over ''F'' are linked. Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic. The fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (algebra)
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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at lea ... [...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] |
U-invariant
In mathematics, the universal invariant or ''u''-invariant of a field describes the structure of quadratic forms over the field. The universal invariant ''u''(''F'') of a field ''F'' is the largest dimension of an anisotropic quadratic space over ''F'', or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that ''u'' is the smallest number such that every form of dimension greater than ''u'' is isotropic, or that every form of dimension at least ''u'' is universal. Examples * For the complex numbers, ''u''(C) = 1. * If ''F'' is quadratically closed then ''u''(''F'') = 1. * The function field of an algebraic curve over an algebraically closed field has ''u'' ≤ 2; this follows from Tsen's theorem that such a field is quasi-algebraically closed.Lam (2005) p.376 * If ''F'' is a non-real global or local field, or more gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division Algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a field. We call ''D'' a division algebra if for any element ''a'' in ''D'' and any non-zero element ''b'' in ''D'' there exists precisely one element ''x'' in ''D'' with ''a'' = ''bx'' and precisely one element ''y'' in ''D'' such that . For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element ''a'' has a multiplicative inverse (i.e. an element ''x'' with ). Associative division algebras The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite- dimensional as a vector space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biquaternion Algebra
In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense. Definition Let ''F'' be a field of characteristic not equal to 2. A ''biquaternion algebra'' over ''F'' is a tensor product of two quaternion algebras.Lam (2005) p.60Szymiczek (1997) p.452 A biquaternion algebra is a central simple algebra of dimension 16 and degree 4 over the base field: it has exponent (order of its Brauer class in the Brauer group of ''F'') equal to 1 or 2. Albert's theorem Let ''A'' = (''a''1,''a''2) and ''B'' = (''b''1,''b''2) be quaternion algebras over ''F''. The Albert form for ''A'', ''B'' is :q = \left\langle\right\rangle \ . It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of ''A'' and ''B''.Knus et al (1991) p.192 The quaternion algebras are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pfister Neighbour
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] |
Brauer Group
Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Brauer (1929–2021), Austrian painter, poet, and actor, father of Timna Brauer * August Brauer (1863-1917), German zoologist * Friedrich Moritz Brauer (1832–1904), Austrian entomologist and museum director * Georg Brauer (1908–2001), German chemist * Ingrid Arndt-Brauer (born 1961), German politician; member of the Bundestag * Jono Brauer (born 1981), Australian Olympic skier * Max Brauer (1887–1973), German politician; First Mayor of Hamburg * Michael Brauer (contemporary), American audio engineer * Rich Brauer (born 1954), American politician from Illinois; state legislator since 2003 * Richard Brauer (1901–1977), German-American mathematician * Richard H. W. Brauer (contemporary), American art museum director; eponym of the Bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of \mathbb_q(T), the field of rational functions in one variable over the finite field with q=p^n elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s. Formal definitions A ''global field'' is one of the following: ;An algebraic number field An algebraic number field ''F'' is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus ''F'' is a field that contains Q and has finite dimension when considered as a vector space over Q. ;The function field of an algebraic curve over a finite field A function field of a variety is t ... [...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] |
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 the quadratic form is anisotropic. More precisely, 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 an anisotropic 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. A quadratic form ''q'' on a finite-dimensional real vector space ''V'' is anisotropic if and only if ''q'' is a defin ... [...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] |
