---
tag: 1ZB13N78XW
kind: concept
name: Characteristic of a field
msc: ["12E99", 13A99]
specializes: [0F6ES3BRYP]
sources:
  - url: https://en.wikipedia.org/wiki/Characteristic_(algebra)
---

## Summary

The characteristic of a field is the least positive number of copies of $1$ that add up to $0$.
It is $0$ when no such number exists.
It is the [[characteristic of the field as a ring|0F6ES3BRYP]], and for a field it is always $0$ or a prime number.
It decides the [[prime subfield|0TNS65QB87]], and many results of field theory depend on it.
For example, the map $a \mapsto a^p$ is a ring homomorphism in characteristic $p$.

## Explanations

### L1 human

Start at $0$ and add $1$ again and again.
In the rational or real numbers you never come back to $0$, so the characteristic is $0$.
In the field $\F_7$ of integers modulo $7$, seven steps bring you back to $0$, so the characteristic is $7$.
A field can never have characteristic $6$, because $2 \cdot 3 = 0$ would make two nonzero elements multiply to $0$.
The [[definition|5ZXRZB41HE]] states this precisely.