---
tag: 7SV4WA92SM
kind: concept
name: Quadratic field
msc: [11R11]
specializes: [MWBDFS2KJ5]
sources:
  - url: https://en.wikipedia.org/wiki/Quadratic_field
same_as:
  wikidata: Q625519
  wikipedia: Quadratic field
---

## Summary

A quadratic field is an [[algebraic number field|MWBDFS2KJ5]] of degree $2$ over $\Q$.
Each one is $\Q(\sqrt{d})$ for a unique square-free integer $d \ne 0, 1$.
Quadratic fields are the simplest number fields beyond $\Q$, and their arithmetic is closely tied to [[binary quadratic forms|SBDSYA037Q]] and to [[Pell's equation|0AF3DANJ76]].

## Explanations

### L1 human

Take the rational numbers and add one square root, such as $\sqrt{2}$ or $\sqrt{-5}$.
The new numbers have the form $a + b\sqrt{d}$ with $a, b$ rational, and they can be added, multiplied and divided like ordinary fractions.
When $d > 0$ the field sits inside the real numbers and is called real; when $d < 0$ it is called imaginary.
Already here unique factorisation can fail: in $\Q(\sqrt{-5})$ one has $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$.