---
tag: SBDSYA037Q
kind: concept
name: Binary quadratic form
msc: ["11E16"]
specializes: [2FQVQCTV88]
sources:
  - url: https://en.wikipedia.org/wiki/Binary_quadratic_form
same_as:
  wikidata: Q864134
  wikipedia: Binary quadratic form
---

## Summary

A binary quadratic form is a [[quadratic form|0RBQS1JJCR]] in two variables, $ax^2 + bxy + cy^2$, usually with integer coefficients.
Its discriminant $b^2 - 4ac$ is unchanged by integral changes of variables of determinant $1$.
Gauss developed the theory of these forms: which integers they represent, how they fall into equivalence classes, and how the classes combine into a group.
This theory is equivalent to the arithmetic of ideals in [[quadratic fields|7SV4WA92SM]], and the number of classes is a [[class number|KTKK9B1X41]].

## Explanations

### L1 human

The forms $x^2 + y^2$ and $x^2 + 2y^2$ are binary quadratic forms.
A classical question is which numbers they produce when $x$ and $y$ run through the integers.
Fermat found that an odd prime is of the form $x^2 + y^2$ exactly when it leaves remainder $1$ on division by $4$.
Changing variables by an integer matrix of determinant $1$ does not change the set of values, so such forms are grouped into classes.