ABSTRACT

In this chapter we introduce one of the basic concepts of algebraic theory of quadratic forms, the Witt group of an arbitrary field. Nonsingular symmetric spaces over a field K are assorted into equivalence classes of similarity relation, which is coarser than the isometry relation. Direct orthogonal sum (DOS) of bilinear spaces induces addition on the set of similarity classes of nonsingular symmetric spaces over the field K and makes it into an Abelian group, the Witt group of the field K. Our ultimate goal is to endow this group with the structure of a commutative ring. In the next chapter we will study tensor product of bilinear spaces which induces a multiplication on the set of similarity classes and eventually leads to the Witt ring of an arbitrary field in Chapter 15.