---
tag: J85FZEQG38
kind: concept
name: Local field
msc: [11S99, 12J25]
specializes: [04C5ZBG1WT]
sources:
  - url: https://en.wikipedia.org/wiki/Local_field
same_as:
  wikidata: Q1868517
  wikipedia: Local field
---

## Summary

A local field is a field with a nontrivial absolute value for which it is locally compact.
The Archimedean local fields are $\R$ and $\C$.
The others are the finite extensions of the fields $\Q_p$ of [[p-adic numbers|BT6T702C28]] and the fields $\F_q((t))$ of Laurent series over finite fields.
They are the completions of [[global fields|634XKY8RZ0]].
Number theory studies a global field one place at a time through its local fields.

## Explanations

### L1 human

A local field is a field where one can measure size, where Cauchy sequences converge, and where closed balls are compact.
The real numbers are the most familiar example.
The $p$-adic numbers $\Q_p$ are another: there, a ball such as $\Z_p$ is compact, like a closed interval.
Studying an equation in all local fields is a first test for whether it has rational solutions.