---
tag: XKEXQDRTGH
kind: concept
name: Function field
msc: [12F20, 14H05]
sources:
  - url: https://en.wikipedia.org/wiki/Function_field_of_an_algebraic_variety
---

## Summary

A function field over a field $k$ is a finitely generated field extension of $k$.
One views it as the field of rational functions on some algebraic variety.
Its [[transcendence degree|D7TR0YWR9H]] over $k$ is the number of variables, which is the dimension of the variety.
Function fields in one variable correspond to algebraic curves, and over finite fields they behave much like number fields.

## Explanations

### L1 human

On the plane, the rational functions are fractions of polynomials in $x$ and $y$.
On a curve such as $y^2 = x^3 + 1$, one uses the same fractions, but $y^2$ and $x^3 + 1$ count as equal.
The field of all such functions is the function field of the curve.
Many questions about a curve, such as whether two curves are birational, become questions about its function field.