Algebraic function field
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In
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
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
The case n = 1 (irreducible algebraic curves in the
The field M(X) of
Number fields and finite fields
The
The study of function fields over a finite field has applications in
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
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
- ISBN 9780817645151.