---
tag: GER28JFXMR
kind: definition
name: Pythagorean field
msc: ["12D15"]
defines: A6M3GTHM40
uses: [0QA8CHH8MG, 778KFSM09P, 02SJG1HC32]
sources:
  - url: https://en.wikipedia.org/wiki/Pythagorean_field
---

## Definition

Let $F$ be a [[field|0QA8CHH8MG]].
The field $F$ is *Pythagorean* if for all $a, b \in F$ there is $c \in F$ with $a^2 + b^2 = c^2$.

## Remarks

By induction on the number of terms, every finite sum of squares in a Pythagorean field is a square.
Example: $\R$ is Pythagorean, since $a^2 + b^2 \ge 0$ has the real square root $\sqrt{a^2 + b^2}$.
More generally, every [[real closed field|778KFSM09P]] and every [[algebraically closed field|02SJG1HC32]] is Pythagorean.
Non-example: $\Q$ is not Pythagorean, because $1^2 + 1^2 = 2$ is not the square of a rational number.
Non-example: $\F_3$ is not Pythagorean, because $1^2 + 1^2 = 2$ and the squares in $\F_3$ are $0$ and $1$.