Editing Rational function (section)

===Notion of a rational function on an algebraic variety=== 
{{Main|Function field of an algebraic variety}}

Like [[Polynomial ring#The polynomial ring in several variables|polynomials]], rational expressions can also be generalized to ''n'' indeterminates ''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>, by taking the field of fractions of ''F''[''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>], which is denoted by ''F''(''X''<sub>1</sub>,..., ''X''<sub>''n''</sub>).

An extended version of the abstract idea of rational function is used in algebraic geometry. There the [[function field of an algebraic variety]] ''V'' is formed as the field of fractions of the [[coordinate ring]] of ''V'' (more accurately said, of a [[Zariski topology|Zariski]]-[[dense subset|dense]] affine open set in ''V''). Its elements ''f'' are considered as regular functions in the sense of algebraic geometry on non-empty open sets ''U'', and also may be seen as morphisms to the [[projective line]].