---
tag: RVEFNEJSHF
kind: definition
name: Global field
msc: [11R04, 11R58]
defines: 634XKY8RZ0
uses: [04C5ZBG1WT, 0C1Y04SZ1W, 0NCJ5WXN6Z, 0ZXZ82FTX2, MWBDFS2KJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Global_field
---

## Definition

A *global field* is a [[field|04C5ZBG1WT]] $K$ of one of the following two kinds.

1. $K$ is a [[finite extension|0C1Y04SZ1W]] of the [[rational numbers|0NCJ5WXN6Z]] $\Q$; then $K$ is a [[number field|MWBDFS2KJ5]].
2. $K$ is a finite extension of the field $\F_q(t)$ of rational functions in one variable $t$ over a [[finite field|0ZXZ82FTX2]] $\F_q$. Then $K$ is a global function field.

## Remarks

Example: $\Q(\sqrt{2})$ is a global field, because it has degree $2$ over $\Q$.
Example: $\F_3(t)$ itself is a global function field, of degree $1$ over $\F_3(t)$.
Non-example: $\R$ is not a global field, because it has infinite degree over $\Q$ and has characteristic $0$.
Non-example: $\C(t)$ is not a global field, because its field of constants $\C$ is not finite.
The global function fields are exactly the function fields of smooth projective curves over finite fields.