Editing Function field (scheme theory) (section)

==Further issues==

Once ''K<sub>X</sub>'' is defined, it is possible to study properties of ''X'' which depend only on ''K<sub>X</sub>''. This is the subject of [[birational geometry]].

If ''X'' is an [[algebraic variety]] over a field ''k'', then over each open set ''U'' we have a [[field extension]] ''K<sub>X</sub>''(''U'') of ''k''. The dimension of ''U'' will be equal to the [[transcendence degree]] of this field extension. All finite transcendence degree field extensions of ''k'' correspond to the rational function field of some 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.