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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 (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 V(''I'') defined by this ideal consists of all ''n''-tuples x = (''x''1,...,''x''''n'') in ''Kn'' such that ''f''(x) = 0 for all ''f'' in ''I''. Hilbert's Nullstell ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rabinowitsch Trick
In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under his original name , is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called ''weak'' Nullstellensatz), by introducing an extra variable. The Rabinowitsch trick goes as follows. Let ''K'' be an algebraically closed field. Suppose the polynomial ''f'' in ''K'' 'x''1,...''x''''n''vanishes whenever all polynomials ''f''1,....,''f''''m'' vanish. Then the polynomials ''f''1,....,''f''''m'', 1 − ''x''0''f'' have no common zeros (where we have introduced a new variable ''x''0), so by the weak Nullstellensatz for ''K'' 'x''0, ..., ''x''''n''they generate the unit ideal of ''K'' 'x''0 ,..., ''x''''n'' Spelt out, this means there are polynomials g_0,g_1,\dots,g_m \in K _0,x_1,\dots,x_n/math> such that : 1 = g_0(x_0,x_1,\dots,x_n) (1 - x_0 f(x_1,\dots,x_n)) + \sum_^m g_i(x_0,x_1,\dots,x_n) f_i(x_1,\dots,x_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field Extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently. Definition and notation Suppose that ''E''/''F'' is a field extension. Then ''E'' may be considered as a vector space over ''F'' (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by :F The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension ''E''/''F'' is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Algebra
In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements ''a''1,...,''a''''n'' of ''A'' such that every element of ''A'' can be expressed as a polynomial in ''a''1,...,''a''''n'', with coefficients in ''K''. Equivalently, there exist elements a_1,\dots,a_n\in A s.t. the evaluation homomorphism at =(a_1,\dots,a_n) :\phi_\colon K _1,\dots,X_ntwoheadrightarrow A is surjective; thus, by applying the first isomorphism theorem, A \simeq K _1,\dots,X_n(\phi_). Conversely, A:= K _1,\dots,X_nI for any ideal I\subset K _1,\dots,X_n/math> is a K-algebra of finite type, indeed any element of A is a polynomial in the cosets a_i:=X_i+I, i=1,\dots,n with coefficients in K. Therefore, we obtain the following characterisation of finitely generated K-algebras :A is a finitely generated K-algebra if and only if it is isomorphic to a quotient ring of the type K _1,\d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski's Lemma
In algebra, Zariski's lemma, proved by , states that, if a field is finitely generated as an associative algebra over another field , then is a finite field extension of (that is, it is also finitely generated as a vector space). An important application of the lemma is a proof of the weak form of Hilbert's nullstellensatz: if ''I'' is a proper ideal of k _1, ..., t_n/math> (''k'' algebraically closed field), then ''I'' has a zero; i.e., there is a point ''x'' in k^n such that f(x) = 0 for all ''f'' in ''I''. (Proof: replacing ''I'' by a maximal ideal \mathfrak, we can assume I = \mathfrak is maximal. Let A = k _1, ..., t_n/math> and \phi: A \to A / \mathfrak be the natural surjection. By the lemma A / \mathfrak is a finite extension. Since ''k'' is algebraically closed that extension must be ''k''. Then for any f \in \mathfrak, :f(\phi(t_1), \cdots, \phi(t_n)) = \phi(f(t_1, \cdots, t_n)) = 0; that is to say, x = (\phi(t_1), \cdots, \phi(t_n)) is a zero of \mathfrak.) The lem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or ''pure existence theorem''), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof. A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics. Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a simple module over ''R''. If ''R'' has a unique maximal right ideal, then ''R'' is known as a local ring, and the maximal right ideal is also the unique maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure Operator
In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are determined by their closed sets, i.e., by the sets of the form cl(''X''), since the closure cl(''X'') of a set ''X'' is the smallest closed set containing ''X''. Such families of "closed sets" are sometimes called closure systems or "Moore families", in honor of E. H. Moore who studied closure operators in his 1910 ''Introduction to a form of general analysis'', whereas the concept of the closure of a subset originated in the work of Frigyes Riesz in connection with topological spaces. Though not formalized at the time, the idea of closure originated in the late 19th century with notable contributions by Ernst Schröder, Richard Dedekind and Georg Cantor. Closure operators are also called "hull operators", which prevents confusion with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Closure
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois. A Galois connection can also be defined on preordered sets or classes; this article presents the common case of posets. The literature contains two closely related notions of "Galois connection". In this article, we will refer to them as (monotone) Galois connections and antitone Galois connections. A Galois connection is rather weak compared to an order isomorphism between the involved posets, but every Galois connection gives rise to an isomorphism of certain sub-posets, as will be explained below. The term Galois correspondence is sometimes used to mean a bijective ''Galois connection' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
