Powers of the Fundamental Ideal in the Witt Ring

Let K be a field of characteristic different from 2. In the algebraic theory of quadratic forms, one studies the Witt ring W(K) of equivalence classes of non-degenerate quadratic forms. The Witt ring has a filtration given by the powers In(K) of the fundamental ideal I(K) of even-dimensional forms. The ideal In(K) is generated by the set Pn(K) of n-fold Pfister forms, a1,…,an ni = 1 1, −ai , where ai K* = K\;{0}. Many questions about this filtration and its quotients have arisen in the study of W(K). The spectacular work of Voevodsky, together with his collaborative work with Orlov and Vishik, allows one to answer many old questions. The purpose of this paper is to indicate some of these solutions.