# Algebraic function field

*https://en.wikipedia.org/wiki/Algebraic_function_field*

In
mathematics, an **algebraic function field** (often abbreviated as **function field**) of *n* variables over the
field *k* is a finitely generated
field extension *K*/*k* which has
transcendence degree *n* over *k*.^{
[1]} Equivalently, an algebraic function field of *n* variables over *k* may be defined as a
finite field extension of the field *K* = *k*(*x*_{1},...,*x*_{n}) of
rational functions in *n* variables over *k*.

## Example

As an example, in the
polynomial ring *k* [*X*,*Y*] consider the
ideal generated by the
irreducible polynomial *Y*^{ 2} − *X*^{ 3} and form the
field of fractions of the
quotient ring *k* [*X*,*Y*]/(*Y*^{ 2} − *X*^{ 3}). This is a function field of one variable over *k*; it can also be written as (with degree 2 over ) or as (with degree 3 over ). We see that the degree of an algebraic function field is not a well-defined notion.

## Category structure

The algebraic function fields over *k* form a
category; the
morphisms from function field *K* to *L* are the
ring homomorphisms *f* : *K* → *L* with *f*(*a*) = *a* for all *a* in *k*. All these morphisms are
injective. If *K* is a function field over *k* of *n* variables, and *L* is a function field in *m* variables, and *n* > *m*, then there are no morphisms from *K* to *L*.

## Function fields arising from varieties, curves and Riemann surfaces

The
function field of an algebraic variety of dimension *n* over *k* is an algebraic function field of *n* variables over *k*.
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 (contravariant equivalence) between the category of varieties over *k* (with
dominant rational maps as morphisms) and the category of algebraic function fields over *k*. (The varieties considered here are to be taken in the
scheme sense; they need not have any *k*-rational points, like the curve *X*^{2} + *Y*^{2} + 1 = 0 defined over the
reals, that is with *k* = **R**.)

The case *n* = 1 (irreducible algebraic curves in the
scheme sense) is especially important, since every function field of one variable over *k* arises as the function field of a uniquely defined
regular (i.e. non-singular) projective irreducible algebraic curve over *k*. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with
dominant
regular maps as morphisms) and the category of function fields of one variable over *k*.

The field M(*X*) of
meromorphic functions defined on a connected
Riemann surface *X* is a function field of one variable over the
complex numbers **C**. In fact, M yields a duality (contravariant equivalence) between the category of compact connected Riemann surfaces (with non-constant
holomorphic maps as morphisms) and function fields of one variable over **C**. A similar correspondence exists between compact connected
Klein surfaces and function fields in one variable over **R**.

## 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. (For example, 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

Given any algebraic function field *K* over *k*, we can consider the
set of elements of *K* which are
algebraic over *k*. These elements form a field, known as the *field of constants* of the algebraic function field.

For instance, **C**(*x*) is a function field of one variable over **R**; its field of constants is **C**.

## Valuations and places

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

Given an algebraic function field *K*/*k* of one variable, we define the notion of a *valuation ring* of *K*/*k*: this is a
subring *O* of *K* that contains *k* and is different from *k* and *K*, and such that for any *x* in *K* we have *x* ∈ *O* or *x*^{ -1} ∈ *O*. Each such valuation ring is a
discrete valuation ring and its maximal ideal is called a *place* of *K*/*k*.

A *discrete valuation* of *K*/*k* is a
surjective function *v* : *K* → **Z**∪{∞} such that *v*(x) = ∞ iff *x* = 0, *v*(*xy*) = *v*(*x*) + *v*(*y*) and *v*(*x* + *y*) ≥ min(*v*(*x*),*v*(*y*)) for all *x*, *y* ∈ *K*, and *v*(*a*) = 0 for all *a* ∈ *k* \ {0}.

There are natural bijective correspondences between the set of valuation rings of *K*/*k*, the set of places of *K*/*k*, and the set of discrete valuations of *K*/*k*. These sets can be given a natural
topological structure: the
Zariski–Riemann space of *K*/*k*.

## See also

- function field of an algebraic variety
- function field (scheme theory)
- algebraic function
- Drinfeld module

## References

**^**Gabriel Daniel & Villa Salvador (2007).*Topics in the Theory of Algebraic Function Fields*. Springer. ISBN 9780817645151.