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. |