  Let F be an arbitrary field of characteristic not 2. We write W(F) for the
Witt ring of F, consisting of the isomorphism classes of all anisotropic
quadratic forms over F. For any element x of W(F), dimension dim x is defined
as the dimension of a quadratic form representing x. The elements of all even
dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the
powers I(F)^n of this ideal plays a fundamental role in the algebraic theory of
quadratic forms. The Milnor conjectures, recently proved by Voevodsky and
Orlov-Vishik-Voevodsky, describe the successive quotients I(F)^n/I(F)^{n+1} of
this filtration, identifying them with Galois cohomology groups and with the
Milnor K-groups modulo 2 of the field F. In the present article we give a
complete answer to a different old-standing question concerning I(F)^n, asking
about the possible values of dim x for x in I(F)^n. More precisely, for any
positive integer n, we prove that the set dim I^n of all dim x for all x in
I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all
even integers greater or equal to 2^{n+1}. Previously available partial
informations on dim I^n include the classical Arason-Pfister theorem, saying
that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's
theorem, saying the same on the integers between 2^n and 2^n+2^{n-1} (the case
n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in
Chow groups of powers of projective quadrics (involving the Steenrod
operations); the method developed can be also applied to other types of
algebraic varieties.
