---
tag: 0QA8CHH8MG
kind: definition
name: Field
msc: ["12E99"]
defines: 04C5ZBG1WT
uses: [0GYGHKX834, 019NFGFPR5]
sources:
  - url: https://en.wikipedia.org/wiki/Field_(mathematics)
    locator: Definition
---

## Definition

A field is a [[commutative ring|0GYGHKX834]] $K$ with these two properties.

1. $1 \neq 0$ in $K$.
2. Every nonzero element of $K$ is a [[unit|019NFGFPR5]]: for each $a \in K$ with $a \neq 0$, there is $b \in K$ with $a b = 1$.

## Remarks

The rings $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are fields. The ring $\mathbb{Z}$ is not a field.
For a prime number $p$, the ring [[$\mathbb{Z}/p\mathbb{Z}$ is a field|0NV6E43P4A]].
Every field is an [[integral domain|09S3PFBEPM]].
A [[division ring|0WZXK05ME7]] has the same axioms without the commutativity of the multiplication.

## Explanations

### L1 human

A field is a number system where the four operations of arithmetic work.
You can add, subtract, multiply, and divide by any number that is not zero.
The rational numbers are the simplest example.