Editing Algebraic function field (section)

==Function fields arising from varieties, curves and Riemann surfaces==
The [[function field of an algebraic variety]] of dimension ''n'' over ''k'' is an algebraic function field of ''n'' variables over ''k''.
Two varieties are [[birational geometry|birationally equivalent]] if and only if their function fields are isomorphic. (But note that non-[[morphism of varieties|isomorphic]] varieties may have the same function field!) Assigning to each variety its function field yields a [[equivalence of categories|duality]] (contravariant equivalence) between the category of varieties over ''k'' (with [[rational mapping|dominant rational maps]] as morphisms) and the category of algebraic function fields over ''k''. (The varieties considered here are to be taken in the [[scheme (mathematics)|scheme]] sense; they need not have any ''k''-rational points, like the curve {{math|1=''X''<sup>2</sup> + ''Y''<sup>2</sup> + 1 = 0}} defined over the [[Real number|reals]], that is with {{math|1=''k'' = '''R'''}}.)

The case ''n''&thinsp;=&thinsp;1 (irreducible algebraic curves in the [[scheme (mathematics)|scheme]] sense) is especially important, since every function field of one variable over ''k'' arises as the function field of a uniquely defined [[regular scheme|regular]] (i.e. non-singular) projective irreducible algebraic curve over ''k''. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with [[Glossary of scheme theory#dominant|dominant]] [[regular map (algebraic geometry)|regular map]]s as morphisms) and the category of function fields of one variable over ''k''.

The field M(''X'') of [[meromorphic function]]s defined on a connected [[Riemann surface]] ''X'' is a function field of one variable over the [[complex number]]s '''C'''. In fact, M yields a duality (contravariant equivalence) between the category of compact connected Riemann surfaces (with non-constant [[holomorphic]] maps as morphisms) and function fields of one variable over '''C'''. A similar correspondence exists between compact connected [[Klein surface]]s and function fields in one variable over '''R'''.