---
tag: 0VPFBAY7MW
kind: definition
name: Prime subfield
msc: ["12E99"]
defines: 0TNS65QB87
uses: [0QA8CHH8MG, 07QY5RPFBR, 0P7CH1CND5]
sources:
  - url: https://en.wikipedia.org/wiki/Field_(mathematics)
    locator: Subfields and prime fields
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]].
The prime subfield of $K$ is the intersection of all the [[subfields|07QY5RPFBR]] of $K$.
It is the smallest subfield of $K$.

## Remarks

If the [[characteristic|0P7CH1CND5]] of $K$ is a prime $p$, then the prime subfield is isomorphic to $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
If the characteristic of $K$ is $0$, then the prime subfield is isomorphic to $\mathbb{Q}$.
The prime subfield of $\mathbb{R}$ is $\mathbb{Q}$.

## Explanations

### L1 human

Every subfield contains $0$ and $1$, and so it contains all the sums $1 + 1 + \dots + 1$ and their quotients.
These elements form the prime subfield.
In the real numbers, they are exactly the rational numbers.