Witt’s chain isometry theorem

We will prove here the second basic result due to Witt, the chain isometry (or piecewise equivalence) theorem. Its importance stems from the fact that, in a sense, it reduces classification of symmetric bilinear spaces up to isometry, to the classification of 2–dimensional spaces. First we discuss chain isometry in terms of orthogonal bases, and then we switch to the more subtle case of diagonalizations, where no restrictions on the characteristic of the underlying field are necessary. The approach to chain isometry presented in this chapter is due to I. Kaplansky [45] and I. Kaplansky and R. J. Shaker [46].