---
tag: 634XKY8RZ0
kind: concept
name: Global field
msc: [11R04, 11R58]
specializes: [04C5ZBG1WT]
sources:
  - url: https://en.wikipedia.org/wiki/Global_field
same_as:
  wikidata: Q1531713
  wikipedia: Global field
---

## Summary

A global field is either a [[number field|MWBDFS2KJ5]] or a global function field.
A number field is a finite extension of $\Q$, and a global function field is a finite extension of $\F_q(t)$ for a finite field $\F_q$.
The two kinds share much of their arithmetic: rings of integers, class groups, the product formula, adeles and [[class field theory|BEKCEYHCHX]].
Their completions at places are the [[local fields|J85FZEQG38]].

## Explanations

### L1 human

The rational numbers $\Q$ and the rational functions $\F_p(t)$ over a finite field behave alike in many ways.
In both, there is a notion of prime, unique factorization in a suitable ring, and many absolute values.
A global field is a finite extension of one of these two basic fields.
For example, $\Q(i)$, the numbers $a + bi$ with $a, b$ rational, is a global field of the first kind.
Studying both kinds side by side often helps: a hard question about numbers can have an easier twin about functions.