Algebraic function field
In mathematics, an algebraic function field of variables over a field is a finitely generated field extension which has transcendence degree over. Equivalently, an algebraic function field of variables over may be defined as a finite field extension of the field of rational functions in variables over.
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. This is a function field of one variable over ; it can also be written as or as . We see that the degree of an algebraic function field is not a well-defined notion.Category structure
The algebraic function fields over form a category; the morphisms from function field to are the ring homomorphisms with for all in. All these morphisms are injective. If is a function field over of variables, and is a function field in variables, and, then there are no morphisms from to.Function fields arising from varieties, curves and Riemann surfaces
The function field of an algebraic variety of dimension over is an algebraic function field of variables over.Two varieties are birationally equivalent if and only if their function fields are isomorphic. Assigning to each variety its function field yields a duality between the category of varieties over and the category of algebraic function fields over. The varieties considered here are to be taken in the scheme sense; they need not have any -rational points, like the curve defined over the real numbers.
The case is especially important, since every function field of one variable over arises as the function field of a uniquely defined regular projective irreducible algebraic curve over. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves and the category of function fields of one variable over.
The field of meromorphic functions defined on a connected Riemann surface is a function field of one variable over the complex numbers. In fact, yields a duality between the category of compact connected Riemann surfaces and function fields of one variable over. A similar correspondence exists between compact connected Klein surfaces and function fields in one variable over.
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. 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 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
Given any algebraic function field over, we can consider the set of elements of which are algebraic over. These elements form a field, known as the field of constants of the algebraic function field.For instance, is a function field of one variable over ; its field of constants is.
Valuations and places
Key tools to study algebraic function fields are absolute values, valuations, places and their completions.Given an algebraic function field of one variable, we define the notion of a valuation ring of : this is a subring of that contains and is different from and, and such that for any in we have or. Each such valuation ring is a discrete valuation ring and its maximal ideal is called a place of.
A discrete valuation of is a surjective function such that for all,
- ,
- ,
There are natural bijective correspondences between the set of valuation rings of, the set of places of, and the set of discrete valuations of. These sets can be given a natural topological structure: the Zariski–Riemann space of.