Function field (scheme theory)

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.