Quadratic Forms and the Orthogonal Group

We leave the arithmetic theory of fields in order to develop a different subject, the abstract theory of quadratic forms. In the latter half of the book we shall combine these two subjects into the arithmetic theory of quadratic forms. Our immediate purpose is to introduce a quadratic form and an orthogonal geometry on an arbitrary finite dimensional vector space and to study certain groups of linear transformations that leave the quadratic form invariant. We must make the assumption from now on that the field of scalars F does not have characteristic 2. As we indicated, our vector spaces are assumed to be finite dimensional.