---
tag: KFVY0RPGF2
kind: concept
name: Witt group
msc: ["11E81", 19G12, 15A63]
sources:
  - url: https://en.wikipedia.org/wiki/Witt_group
same_as:
  wikidata: Q3117708
  wikipedia: Witt group
---

## Summary

The Witt group of a field $F$ of characteristic not $2$ is an abelian group built from the nondegenerate [[quadratic forms|0RBQS1JJCR]] over $F$.
Two forms give the same element when they differ only by hyperbolic planes, and the group operation is the orthogonal sum.
It collects the essential information about quadratic forms over $F$ into one algebraic object.
With the tensor product of forms it becomes a ring, the Witt ring.

## Explanations

### L1 human

Quadratic forms can be added by placing them side by side in separate variables.
Some forms, the hyperbolic ones such as $x^2 - y^2$, carry no interesting information, so they are treated as zero.
After this, every form has a negative, namely the form with all signs reversed, and one gets a group.
For the real numbers, the class of a form is given by its signature, the number of plus signs minus the number of minus signs.
So the Witt group of $\R$ is $\Z$.