Index: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Home > Function field

In algebraic geometry, the function field of an irreducible algebraic variety

is the field of fractions of the ring of regular functions.

The ring of regular functions on a variety V defined over a field K is an integral domain if and only if the variety is irreducible, and in this case the field of fractions is defined. It is a field extension of the ground field K; its transcendence degree is by definition the dimension of the variety. All extensions of K that are finitely-generated as fields arise in this way from some algebraic variety.

In the particular case of an algebraic curve C, that is, dimension 1, it follows that any two non-constant functions F and G on C satisfy a polynomial equation P(F,G) = 0.

Properties of the variety V that depend only on the function field are studied in birational geometry.

Algebraic geometry

Read more »

<Hide>

Topics: Function Field, Algebraically Closed Field, Algebraic Number...

Algebraically closed fieldAbstract algebra In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F has a zero in F''. In that case, every such polynomial splits into linear factors. It can be shown that a field	Algebraic numberAbstract algebra Algebra In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form : a x n + a x n minus;1 + ··· + a x + a 0 where n is a posit	Algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations . When there is more than o
Algebraic extensionAbstract algebra Algebra In abstract algebra, a field extension L ''K is called algebraic if every element of L is algebraic over K i. if every element of L is a root of some non-zero polynomial with coefficients in K''. Field extensions which are not alg	Algebraic closureField theory In mathematics, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that	Algebraic topologyTopology Algebraic topology Abstract algebra Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants The goal is to take topological spaces, and further ca
Algebraic chess notation	Algebraic integer	Algebraic structure
Algebraic element	Algebraic enumeration	Algebraic number field
Algebraic group	Algebraic variety	Algebraic form

Non User