In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex 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.
Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra.
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.
is an integral scheme, then for every open affine subset
Furthermore, it can be verified that these are all the same, and are all equal to the stalk of the generic point of
This point of view is developed further in function field (scheme theory).
All extensions of K that are finitely generated as fields over K arise in this way from some algebraic variety.
This is also the function field of the projective line.
Consider the affine algebraic plane curve defined by the equation