---
tag: W168DS966V
kind: concept
name: Completion of a valued field
msc: [12J10, 13J10]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_field
---

## Summary

The completion of a field with an absolute value is a larger such field, see the [[definition|KFYW0WN9AD]].
In it every Cauchy sequence converges, and the original field is dense.
It is the [[completion of the metric space|06RW4RKWH7]] of the field, with the field operations extended by continuity.
The real numbers are the completion of $\Q$ for the usual absolute value, and the [[p-adic numbers|BT6T702C28]] are its completion for the [[p-adic absolute value|WGN82TCEF2]].
Many problems over a valued field are solved first in its completion, where limits exist and [[Hensel's lemma|010QVB71T2]] applies.

## Explanations

### L1 human

A field such as $\Q$ has holes: the decimal approximations $1, 1.4, 1.41, 1.414, \ldots$ of $\sqrt 2$ get closer and closer to each other but have no limit in $\Q$.
Completing means adding a limit for every such sequence, and doing arithmetic with these limits.
Which sequences count as "getting closer" depends on the absolute value.
For the $5$-adic absolute value, the integers $1, 1 + 5, 1 + 5 + 25, \ldots$ get closer, and their limit in the completion is $-1/4$, because $1 + 5 + 25 + \cdots = 1/(1 - 5)$ in this sense.