---
tag: 7ABT36371X
kind: definition
name: Witt group
msc: ["11E81", 19G12, 15A63]
defines: KFVY0RPGF2
uses: [0RBQS1JJCR, 0F6ES3BRYP, 082GHNPJ53, CCB8VNQCV0]
sources:
  - url: https://en.wikipedia.org/wiki/Witt_group
---

## Definition

Let $F$ be a field whose [[characteristic|0F6ES3BRYP]] is not $2$.
A *quadratic space* over $F$ is a finite-dimensional vector space with a nondegenerate [[quadratic form|0RBQS1JJCR]].
The *hyperbolic plane* $H$ is $F^2$ with the form $q(x, y) = xy$.
For quadratic spaces $q_1$ and $q_2$, write $q_1 \perp q_2$ for their orthogonal sum, the form $(v_1, v_2) \mapsto q_1(v_1) + q_2(v_2)$ on the direct sum.
Write $mH$ for the orthogonal sum of $m$ copies of $H$.
Two quadratic spaces $q_1, q_2$ are *Witt equivalent* when $q_1 \perp mH$ and $q_2 \perp nH$ are isometric for some integers $m, n \ge 0$.
The *Witt group* $W(F)$ is the set of Witt equivalence classes of quadratic spaces over $F$, with the operation $[q_1] + [q_2] = [q_1 \perp q_2]$.

## Remarks

This operation is well defined, and $W(F)$ is an [[abelian group|082GHNPJ53]].
The zero is the class of $H$ and of the zero space, and $-[q] = [-q]$, because $q \perp (-q)$ is a sum of hyperbolic planes.
[[Witt's theorem|CCB8VNQCV0]] gives that each class contains exactly one anisotropic form up to isometry.
Example: $W(\C) \cong \Z/2\Z$, given by the dimension modulo $2$, because over $\C$ the form $x^2 + y^2$ is hyperbolic.
Example: $W(\R) \cong \Z$, given by the signature.
Non-example: the class of $x^2$ in $W(\R)$ is not zero, because $x^2$ is anisotropic and the zero class contains only the zero space as anisotropic form.
With the tensor product of forms, $W(F)$ is a commutative ring, the Witt ring.