---
tag: A6M3GTHM40
kind: concept
name: Pythagorean field
msc: ["12D15"]
sources:
  - url: https://en.wikipedia.org/wiki/Pythagorean_field
same_as:
  wikidata: Q2120067
  wikipedia: Pythagorean field
---

## Summary

A Pythagorean field is a [[field|04C5ZBG1WT]] in which every sum of two squares is again a square.
The name comes from the [[Pythagorean theorem|0VP3A1290K]].
In such a field, a right triangle with legs in the field has a hypotenuse in the field.
Real closed fields and algebraically closed fields are Pythagorean, but $\Q$ is not.

## Explanations

### L1 human

In the rational numbers, the right triangle with legs $1$ and $1$ has hypotenuse $\sqrt 2$, which is not rational.
A Pythagorean field is a field where this never happens: $\sqrt{a^2 + b^2}$ always lies in the field.
The real numbers have this property, and so do the complex numbers.