Function Field (other)

	Function Field (other) Function field may refer to: Function field of an algebraic variety Function field (scheme theory) Algebraic function field Function field sieve Function field analogy This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of pro ... {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Function Field Of An Algebraic Variety In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects which are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions. Definition for complex manifolds In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in \mathbb\cup\infty.) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra. For the Riemann sphere, which is the variety \mathbb^1 over the complex numbers, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Function Field (scheme Theory) The sheaf of rational functions ''KX'' of a scheme ''X'' is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of varieties, such a sheaf associates to each open set ''U'' the ring of all rational functions on that open set; in other words, ''KX''(''U'') is the set of fractions of regular functions on ''U''. Despite its name, ''KX'' does not always give a field for a general scheme ''X''. Simple cases In the simplest cases, the definition of ''KX'' is straightforward. If ''X'' is an (irreducible) affine algebraic variety, and if ''U'' is an open subset of ''X'', then ''KX''(''U'') will be the field of fractions of the ring of regular functions on ''U''. Because ''X'' is affine, the ring of regular functions on ''U'' will be a localization of the global sections of ''X'', and consequently ''KX'' will be the constant sheaf whose value is the fraction field of the global sections of ''X''. If ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Algebraic Function Field In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebraic function field of ''n'' variables over ''k'' may be defined as a finite field extension of the field ''K'' = ''k''(''x''1,...,''x''''n'') of rational functions in ''n'' variables over ''k''. Example As an example, in the polynomial ring ''k'' 'X'',''Y''consider the ideal generated by the irreducible polynomial ''Y''2 − ''X''3 and form the field of fractions of the quotient ring ''k'' 'X'',''Y''(''Y''2 − ''X''3). This is a function field of one variable over ''k''; it can also be written as k(X)(\sqrt) (with degree 2 over k(X)) or as k(Y)(\sqrt (with degree 3 over k(Y)). We see that the degree of an algebraic function field is not a well-defined notion. Category structure The algebraic function fields over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Function Field Sieve In mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic subexponential complexity. Leonard Adleman developed it in 1994 and then elaborated it together with M. D. Huang in 1999.L. Adleman, M.D. Huang. "Function Field Sieve Method for Discrete Logarithms over Finite Fields". In: Inf. Comput. 151 (May 1999), pp. 5-16. DOI: 10.1006/inco.1998.2761. Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two. The discrete logarithm problem in a finite field consists of solving the equation a^x = b for a,b \in \mathbb_ , p a prime number and n an integer. The function f: \mathbb_ \to \mathbb_, x \mapsto a^x for a fixed a \in \mathbb_ is a one-way function used in cryptography. Several cryptographic methods are based on the DLP such as the Diffie-Hellman key exchange, the El Gamal cryptosystem and the Digital Signature Algorithm. Num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]