---
tag: BT5V3TSNY6
kind: concept
name: Place (number field)
msc: [11R04, 12J10]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_number_theory
---

## Summary

A place of an [[algebraic number field|MWBDFS2KJ5]] $K$ is a class of equivalent nontrivial absolute values on $K$.
The finite places correspond to the nonzero prime ideals of the ring of integers.
The infinite places correspond to the real embeddings and the pairs of complex conjugate embeddings of $K$.
Each place gives a completion of $K$, which is a [[local field|J85FZEQG38]] or $\R$ or $\C$.
The places together are the index set of the [[adele ring|SEH68BA1QQ]].
For $K = \Q$, [[Ostrowski's theorem|GCQ23XT6VD]] says that the places are the primes $p$ and one infinite place.

## Explanations

### L1 human

A place is a way of measuring the size of the numbers of a field.
On $\Q$ there is the usual size, and for each prime $p$ there is the [[p-adic size|WGN82TCEF2]].
For the $p$-adic size, a number is small when it is divisible by a high power of $p$.
For example, $\abs{75}_5 = 1/25$, because $75 = 3 \cdot 5^2$.
Number theorists treat all these ways of measuring on an equal footing, and the places are their list.