The ring of regular functions on a variety V defined over a field Kis 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.
Properties of the variety V that depend only on the function field are studied in birational geometry.