# 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.

## Definition for complex manifolds

In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in $\mathbb {C} \cup \infty$ .) Together with the operations of addition and multiplication of functions, this is a field in the sense of algebra.

For the Riemann sphere, which is the variety $\mathbb {P} ^{1}$ over the complex numbers, the global meromorphic functions are exactly the rational functions (that is, the ratios of complex polynomial functions).

## Construction in algebraic geometry

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.

## Generalization to arbitrary scheme

In the most general setting, that of modern scheme theory, we take the latter point of view above as a point of departure. Namely, if $X$ is an integral scheme, then for every open affine subset $U$ of $X$ the ring of sections ${\mathcal {O}}_{X}(U)$ on $U$ is an integral domain and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the local ring of the generic point of $X$ . Thus the function field of $X$ is just the local ring of its generic point. This point of view is developed further in function field (scheme theory). See Robin Hartshorne (1977).

## Geometry of the function field

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.

## Examples

The function field of a point over K is K.

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.

Consider the affine plane curve defined by the equation $y^{2}=x^{5}+1$ . Its function field is the field K(x,y), generated by elements x and y that are transcendental over K and satisfy the algebraic relation $y^{2}=x^{5}+1$ .