Witt's Decomposition Theorem

Get Witt's Decomposition Theorem essential facts below. View Videos or join the Witt's Decomposition Theorem discussion. Add Witt's Decomposition Theorem to your PopFlock.com topic list for future reference or share this resource on social media.
## Statement

## Witt's cancellation theorem

## Witt's decomposition theorem

## Citations

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Witt's Decomposition Theorem

*"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki-Witt fixed point theorem of order theory.*

In mathematics, **Witt's theorem**, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field *k* may be extended to an isometry of the whole space. An analogous statement holds also for skew-symmetric, Hermitian and skew-Hermitian bilinear forms over arbitrary fields. The theorem applies to classification of quadratic forms over *k* and in particular allows one to define the Witt group *W*(*k*) which describes the "stable" theory of quadratic forms over the field *k*.

Let be a finite-dimensional vector space over a field *k* of characteristic different from 2 together with a non-degenerate symmetric or skew-symmetric bilinear form. If is an isometry between two subspaces of *V* then *f* extends to an isometry of *V*.

Witt's theorem implies that the dimension of a maximal totally isotropic subspace (null space) of *V* is an invariant, called the **index** or **Witt index** of *b*,^{[1]} and moreover, that the isometry group of acts transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and representation theory of the isometry group and in the theory of reductive dual pairs.

Let , , be three quadratic spaces over a field *k*. Assume that

Then the quadratic spaces and are isometric:

In other words, the direct summand appearing in both sides of an isomorphism between quadratic spaces may be "cancelled".

Let be a quadratic space over a field *k*. Then
it admits a **Witt decomposition**:

where is the radical of *q*, is an anisotropic quadratic space and is a split quadratic space. Moreover, the anisotropic summand, termed the **core form**, and the hyperbolic summand in a Witt decomposition of are determined uniquely up to isomorphism.^{[2]}

Quadratic forms with the same core form are said to be *similar* or **Witt equivalent**.

**^**Lam 2005, p. 12.**^**Lorenz 2008, p. 30.

- Emil Artin (1957) Geometric Algebra, page 121
- Lam, Tsit-Yuen (2005),
*Introduction to Quadratic Forms over Fields*, Graduate Studies in Mathematics,**67**, American Mathematical Society, ISBN 0-8218-1095-2, MR 2104929, Zbl 1068.11023 - Lorenz, Falko (2008),
*Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics*, Springer-Verlag, pp. 15-27, ISBN 978-0-387-72487-4, Zbl 1130.12001 - O'Meara, O. Timothy (1973),
*Introduction to Quadratic Forms*, Die Grundlehren der mathematischen Wissenschaften,**117**, Springer-Verlag, Zbl 0259.10018

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists