Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.

In classical algebraic geometry, we generalize the second point of view. For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an affine coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety V, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data as agree on the intersections of open affines. We may define the function field of V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.

If V is a variety defined over a field K, then the function field K(V) is a finitely generated field extension of the ground field K; its transcendence degree is equal to the dimension of the variety. All extensions of K that are finitely-generated as fields over K arise in this way from some algebraic variety. These field extensions are also known as algebraic function fields over K.

Properties of the variety V that depend only on the function field are studied in birational geometry.

The function field of the affine line over K is isomorphic to the field K(t) of rational functions in one variable. This is also the function field of the projective line.