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Residue Field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ring and \mathfrak is then its unique maximal ideal. In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the ''natural domain'' for the coordinates of the point. Definition Suppose that R is a commutative local ring, with maximal ideal \mathfrak. Then the residue field is the quotient ring R/\mathfrak. Now suppose that X is a scheme and x is a point of X. By the definition of a scheme, we may find an affine neighbourhood \mathcal = \text(A) of x, with some commutative ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Domain
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element ''a'' has the cancellation property, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using " domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entire ring for integral domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Zeta Function
In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most-fundamental objects of number theory. Definition The arithmetic zeta function is defined by an Euler product analogous to the Riemann zeta function: : = \prod_ \frac, where the product is taken over all closed points of the scheme . Equivalently, the product is over all points whose residue field is finite. The cardinality of this field is denoted . Examples and properties Varieties over a finite field If is the spectrum of a finite field with elements, then :\zeta_X(s) = \frac. For a variety ''X'' over a finite field, it is known by Grothendieck's trace formula that :\zeta_X(s) = Z(X, q^) where Z(X, t) is a rational function (i.e., a quotient of polynomials). Given two varieties ''X'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Of A Scheme
In algebraic geometry, the dimension of a scheme is a generalization of a dimension of an algebraic variety. Scheme theory emphasizes the relative point of view and, accordingly, the relative dimension of a morphism of schemes is also important. Definition By definition, the dimension of a scheme ''X'' is the dimension of the underlying topological space: the supremum of the lengths ''ℓ'' of chains of irreducible closed subsets: :\emptyset \ne V_0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_\ell \subset X. In particular, if X = \operatorname A is an affine scheme, then such chains correspond to chains of prime ideals (inclusion reversed) and so the dimension of ''X'' is precisely the Krull dimension of ''A''. If ''Y'' is an irreducible closed subset of a scheme ''X'', then the codimension of ''Y'' in ''X'' is the supremum of the lengths ''ℓ'' of chains of irreducible closed subsets: :Y = V_0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_\ell \subset X. An irreducible subse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ''L''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains the multiplicative identity 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism Of Schemes
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes. A morphism of algebraic stacks generalizes a morphism of schemes. Definition By definition, a morphism of schemes is just a morphism of locally ringed spaces. Isomorphisms are defined accordingly. A scheme, by definition, has open affine charts and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties). Let ƒ:''X''→''Y'' be a morphism of schemes. If ''x'' is a point of ''X'', since ƒ is continuous, there are open affine subsets ''U'' = Spec ''A'' of ''X'' containing ''x'' and ''V'' = Spec ''B'' of ''Y'' such that ƒ(''U'') ⊆ ''V''. Then ƒ: ''U'' → ''V'' is a morphism of affine schemes and thus is induced by some ring homomorphism ''B'' → ''A'' (cf. #Affine case.) In fact, one can use this des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendence Degree
In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients in K. In other words, a transcendental extension is a field extension that is not algebraic. For example, \mathbb and \mathbb are both transcendental extensions of \mathbb. A transcendence basis of a field extension L/K (or a transcendence basis of L over K) is a maximal algebraically independent subset of L over K. Transcendence bases share many properties with bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero. Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic varie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is a point in a ''general position'', at which all generic property, generic properties are true, a generic property being a property which is true for Almost everywhere, almost every point. In classical algebraic geometry, a generic point of an affine algebraic variety, affine or projective algebraic variety of dimension ''d'' is a point such that the field generated by its coordinates has transcendence degree ''d'' over the field generated by the coefficients of the equations of the variety. In scheme theory, the spectrum of a ring, spectrum of an integral domain has a unique generic point, which is the zero ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space ''X'' is a point whose closure is ''X''. Definition and motivation A generic point of the topological space ''X'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem). Formulation Let k be a field (such as the rational numbers) and K be an algebraically closed field extension of k (such as the complex numbers). Consider the polynomial ring k _1, \ldots, X_n/math> and let I be an ideal in this ring. The algebraic set \mathrm V(I) defined by this ideal consists of all n-tuples \mathbf x = (x_1, \dots, x_n) in K^n such that f(\mathbf x) = 0 for all f in Hilbert's Nullstellensatz states that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism Of Finite Type
In commutative algebra, given a homomorphism A\to B of commutative rings, B is called an A-algebra of finite type if B can be finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means that B is finitely generated as an A- module. For example, for any commutative ring A and natural number n, the polynomial ring A _1,\dots,x_n/math> is an A-algebra of finite type, but it is not a finite A-algebra unless A = 0 or n = 0. Another example of a finite-type homomorphism that is not finite is \mathbb \to \mathbb x,y]/(y^2 - x^3 - t). The analogous notion in terms of scheme (mathematics), schemes is that a morphism f:X\to Y of schemes is of finite type if Y has a covering by affine open subschemes V_i=\operatorname(A_i) such that f^(V_i) has a finite covering by affine open subschemes U_=\operatorname(B_) of X with B_ an A_i-algebra of finite type. One also says that X is of finite type over Y. For example, for any natural number n and field k, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as \left(x - \sqrt\right)\left(x + \sqrt\right) if it is considered as a polynomial with real coefficients. One says that the polynomial is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a pol ... [...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 . In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. Every field K is contained in an algebraically closed field C, and the roots in C of the polynomials with coefficients in K form an algebraically closed field called an algebraic closure of K. Given two algebraic closures of K there are isomorphisms between them that fix the elements of K. Algebraically closed fields appear in the following chain of class inclusions: 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 cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
