In mathematics, an algebraic function field (often abbreviated as function field variables over a field is a finitely generated field extension which has transcendence degree. may be defined as a finite field extension of rational functions.

Example

As an example, in the polynomial ring consider the ideal generated by the irreducible polynomial and form the field of fractions of the quotient ring). We see that the degree of an algebraic function field is not a well-defined notion.

Category structure

form a category; the morphisms are the ring homomorphisms. All these morphisms are injective.

Function fields arising from varieties, curves and Riemann surfaces

The function field of an algebraic variety. Two varieties are birationally equivalent if and only if their function fields are isomorphic (but note that non-isomorphic varieties may have the same function field). Assigning to each variety its function field yields a duality (with dominant rational maps. The varieties considered here are to be taken in the scheme defined over the real numbers.

(irreducible algebraic curves in the scheme arises as the function field of a uniquely defined regular. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with dominant regular maps.

of meromorphic functions defined on a connected Riemann surface is a function field of one variable over the complex numbers yields a duality between the category of compact connected Riemann surfaces (with non-constant holomorphic. A similar correspondence exists between compact connected Klein surfaces.

Number fields and finite fields

The function field analogy states that almost all theorems on number fields have a counterpart on function fields of one variable over a finite field, and these counterparts are frequently easier to prove (see analogue for irreducible polynomials over a finite field). In the context of this analogy, both number fields and function fields over finite fields are usually called "global fields".

The study of function fields over a finite field has applications in cryptography and error correcting codes. For example, the function field of an elliptic curve over a finite field (an important mathematical tool for public key cryptography) is an algebraic function field.

Function fields over the field of rational numbers play also an important role in solving inverse Galois problems.

Field of constants

, we can consider the set which are algebraic. These elements form a field, known as the field of constants of the algebraic function field.

.

Valuations and places

Key tools to study algebraic function fields are absolute values, valuations, places and their completions.

of one variable, we define the notion of a valuation ring: this is a subring. Each such valuation ring is a discrete valuation ring and its maximal ideal is called a place.

A discrete valuation is a surjective,

• ,

• ,

.

. These sets can be given a natural topological structure: the Zariski–Riemann space.

See also

• function field of an algebraic variety

• function field (scheme theory)

• algebraic function

• Drinfeld module

References