Funktionsfält

Funktionsfält, or function field, is a central concept in algebraic geometry and number theory. Let k be a base field and X an integral variety over k. The function field k(X) consists of rational functions on X, i.e., equivalence classes of regular functions on nonempty open subsets, or more formally the field of fractions of the coordinate ring of any nonempty affine open subset. This construction is independent of the chosen open subset and captures the algebraic relations among functions on X.

The field k(X) is a finitely generated field extension of k, and its transcendence degree equals the dimension of X. In particular, for curves (dimension one) the function field is an algebraic function field in one variable over k. Examples include the projective line P^1 over k, whose function field is k(t), the field of rational functions in one indeterminate t; and, for an elliptic curve E given by a Weierstrass equation y^2 = x^3 + ax + b, the function field is k(E) = k(x, y) with the relation y^2 = x^3 + ax + b.

Constants and valuations: the subfield of elements of k(X) that are algebraic over k is called the constant field; in many situations it equals k. Over the complex numbers, the function field of a projective curve corresponds to the field of meromorphic functions on the associated Riemann surface.

Applications: function fields provide a framework for studying maps between varieties via field extensions, as well as valuations (places), genus, and rational points, linking geometric properties to algebraic structures.