---
tag: 2D5SS6WM14
kind: concept
name: Function field of a variety
msc: [14A10, 12F20]
specializes: [XKEXQDRTGH]
sources:
  - url: https://en.wikipedia.org/wiki/Function_field_of_an_algebraic_variety
same_as:
  wikidata: Q5508773
  wikipedia: Function field of an algebraic variety
---

## Summary

The function field $k(X)$ of an algebraic variety $X$ over $k$ is the field of rational functions on $X$.
A rational function is a function regular on some nonempty open subset.
Two of them are identified when they agree on a smaller open set.
For an affine variety it is the field of fractions of the coordinate ring.
It is a finitely generated field extension of $k$.
It is a birational invariant: two varieties have isomorphic function fields over $k$ exactly when they are birational.

## Explanations

### L1 human

A rational function is a fraction of polynomials, like $x/y$, which makes sense except where the denominator vanishes.
On a variety, such a fraction is defined on a large open piece, and we do not care which piece.
The function field collects all of these fractions, and it is a field because every nonzero fraction can be turned upside down.
The function field of the affine line is $k(t)$, the field of rational functions in one variable.