---
tag: X32TSWJSYT
kind: concept
name: Formally real field
msc: ["12D15", 12J15]
sources:
  - url: https://en.wikipedia.org/wiki/Formally_real_field
same_as:
  wikidata: Q17097900
  wikipedia: Formally real field
---

## Summary

A formally real field is a [[field|04C5ZBG1WT]] in which $-1$ is not a sum of squares.
By a theorem of Artin and Schreier, these are exactly the fields that admit an [[ordering|0P3VCBDEE2]].
They are the setting of real algebra, which studies the real numbers through algebraic properties alone.

## Explanations

### L1 human

In the real numbers, a sum of squares is never negative, so it is never $-1$.
A formally real field is a field with this one property of the real numbers.
The rational numbers and the field $\R(t)$ of real rational functions are formally real.
The complex numbers are not, because $-1 = i^2$.