This appendix presents the theory of bilinear forms and quadratic forms which we shall require in Chapters 2 and 3. While the reader has undoubtedly met these topics in a course on linear algebra, the first two sections review them from, perhaps, a more abstract point of view. The abstraction facilitates proofs, in the third section, of Witt's Extension Theorem and Cancellation Theorem for quadratic forms over an arbitrary field (including the case of fields of characteristic 2.) The last section classifies nondegenerate forms over Fp (p odd).

BILINEAR FORMS

Let V be a vector space of finite dimension n over a field F. A bilinear form on V is a map B: V x V→F which is F-linear in each coordinate.

(1) B is said to be a symmetric bilinear form if B(u,v) = B(v,u) for all u, vεV. (B is also said to be an orthogonal bilinear form or a scalar product.)

(2) B is said to be an alternating bilinear form if B(u,u) = 0 for all u V or, equivalently, if B(u,v) = -B(v,u) for all u, vεV. (B is also said to be symplectic.)

In this monograph we are only concerned with symmetric bilinear forms. (Note that an alternating form is symmetric if char F=2.) A pair (V,B), where B is a symmetric bilinear form on the vector space V, is called a scalar product space.