Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety V is the field (mathematics) of fraction (mathematics)s of the ring (mathematics) of regular functions on V .

The ring of regular functions on 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 an algebraic variety 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.