Witt's theorem

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

Witt's theorem, named after Ernst Witt, concerns symmetric bilinear forms on finite-dimensional vector spaces. It tells us when we can extend an isometry on subspaces to an isometry of the whole space. This theorem gives some very powerful results, and allows one to define the Witt group, an invariant of the field you are working over. Fix a field k. A symmetric bilinear form is a pair (E, b), where E is a finite-dimensional vector space over k, and b : E × E → k is a symmetric bilinear map. That is, for all α ∈ k and for all x, y, z ∈ E we have

In different contexts it is convenient to refer to the form simply as E or simply as b. Moreover, all forms here will be bilinear and symmetric, so those adjectives are understood to be there when we simply refer to a form. The maps between forms that we are interested in are metric linear maps

<math> \sigma \colon (E,b) \to (E',b') \,</math>,

where σ is a linear map of vector spaces E → E′ which is metric, which means it preserves the bilinear form

<math> b'(\sigma(x), \sigma(y)) = b(x,y) \,</math> for all <math> x,y \in E. \,</math>

An isometry of forms is an isometry on the vector spaces which is also metric. Clearly, if σ is an isometry of vector spaces which is metric, then σ ⁻¹ is also metric. In addition, we define the kernel of b to be

<math> \ker(b) = \{ x \in E \mid b(x,y) = 0 \ \forall \ y \in E \}. \,</math>

If ker(b) = 0, we say b is non-degenerate.

Witt's theorem

Let (E, b) be a symmetric, non-degenerate bilinear form, then any isometry

<math>\sigma \colon F \to F' \,</math>

of subspaces F, F′ of E can be extended to an isometry E → E. From this, we immediately get the following corollary. However, we must first make the following definition. A subspace W of E is called a nullspace if b(x, y) = 0 for all <math> x,y \in W </math>. A maximal nullspace is a nullspace which is not contained in any larger nullspace. Corollary Let (E, b) be a symmetric, non-degenerate bilinear form, then every maximal nullspace has the same dimension, which is the maximum dimension of any nullspace. This maximum dimension is called the index of E, denoted ind(E). Proof of corollary. Let W be a maximal nullspace, and let V be any nullspace. Suppose dim(W) <math> \leq \,</math> dim(V), then there is a vector space isomorphism from W to a subspace of V. Then, because, both spaces are nullspaces, this map of vector spaces is automatically metric. By Witt's theorem, we can extend this isometry to <math> \sigma \colon E \to E </math>. Thus, <math> \sigma^{-1}(V) </math> is a nullspace containing W. By maximality, <math> W = \sigma^{-1}(V) </math>. Therefore, dim(W) = dim(V).

References

O. Timonthy O'Meara, Introduction to Quadratic Forms, Springer-Verlag, 1973 (formulation and proof of theorem on page 97-99–).