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Hasse–Minkowski Theorem
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent ''locally at all places'', i.e. equivalent over every topological completion (ring theory), completion of the field (which may be real number, real, complex number, complex, or p-adic number, p-adic). A related result is that a quadratic space over a number field is isotropic quadratic form, isotropic if and only if it is isotropic locally everywhere, or equivalently, that a quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field. The theorem was proved in the case of the field of rational numbers by Hermann Minkowski and generalized to number fields by Helmut Hasse. The same statement holds even more generally for all global fields. Importance The importance of the Hasse–Minkowski theorem lies in the novel paradigm it presented for ans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion (ring Theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions ''R'' on a space ''X'' concentrates on a formal neighborhood of a point of ''X'': heuristically, this is a neighborhood so small that ''all'' Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when ''R'' has a metric given by a non-Archimedean absolute value. General construction Suppose that ''E'' is an abelian group with a descending filtration : E = F^0 E \supset F^1 E \supset F^2 E \supset \cdots \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hensel's Lemma
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to a unique root modulo any higher power of . More generally, if a polynomial factors modulo into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of (the case of roots corresponds to the case of degree for one of the factors). By passing to the "limit" (in fact this is an inverse limit) when the power of tends to infinity, it follows that a root or a factorization modulo can be lifted to a root or a factorization over the p-adic integer, -adic integers. These results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where is replaced by an ideal (ring theory), ideal, and "coprime polynomials" means "polynomials that gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Forms
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'' . Over the reals, a quadratic form is said to be '' definite'' if it takes the value zero only when all its variables are simultaneously zero; otherwise it is ''isotropic''. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology ( intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form -\mathbf^\mathsf\boldsy ... [...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] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbach (1997, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Invariant Of A Quadratic Form
In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form ''Q'' over a field ''K'' takes values in the Brauer group Br(''K''). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt. The quadratic form ''Q'' may be taken as a diagonal form :Σ ''a''''i''''x''''i''2. Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras :(''a''''i'', ''a''''j'') for ''i'' < ''j''. This is independent of the diagonal form chosen to compute it.Lam (2005) p.118 It may also be viewed as the second Stiefel–Whitney class of ''Q''. Symbols The invariant may be computed for a specific φ taking values in the group C[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, every element of is a root of a non-zero polynomial with coefficients in . A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic. The algebraic extensions of the field \Q of the rational numbers are called algebraic number fields and are the main objects of study of algebraic number theory. Another example of a common algebraic extension is the extension \Complex/\R of the real numbers by the complex numbers. Some properties All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester's Law Of Inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if A is a symmetric matrix, then for any invertible matrix S, the number of positive, negative and zero eigenvalues (called the inertia of the matrix) of D=SAS^\mathrm is constant. This result is particularly useful when D is diagonal, as the inertia of a diagonal matrix can easily be obtained by looking at the sign of its diagonal elements. This property is named after James Joseph Sylvester who published its proof in 1852. Statement Let A be a symmetric square matrix of order n with real entries. Any non-singular matrix S of the same size is said to transform A into another symmetric matrix , also of order , where S^\mathrm is the transpose of . It is also said that matrices A and B are congruent. If A is the coefficient matrix of some quadratic form of , then B is the matrix fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Place (mathematics)
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and the sum of their squares, equal two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Class
In mathematics, specifically abstract algebra, a square class of a field F is an element of the square class group, the quotient group F^\times/ F^ of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square class is a subset of the nonzero elements (a coset of the multiplicative group) consisting of the elements of the form ''xy''2 where ''x'' is some particular fixed element and ''y'' ranges over all nonzero field elements.. For instance, if F=\mathbb, the field of real numbers, then F^\times is just the group of all nonzero real numbers (with the multiplication operation) and F^ is the subgroup of positive numbers (as every positive number has a real square root). The quotient of these two groups is a group with two elements, corresponding to two cosets: the set of positive numbers and the set of negative numbers. Thus, the real numbers have two square classes, the positive numbers and the negative numbers. Square classes ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discriminant Of A Quadratic Form
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 ... [...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 metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact topological field with respect to a non-discrete topology. The real numbers R, and the complex numbers C (with their standard topologies) are Archimedean local fields. 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 least 250 years, the first examples of non-Archimedean local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
