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Abstract

Let F be an arbitrary field of characteristic ≠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 ∈ W ( F ) , its dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted by 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 longstanding question concerning I ( F ) n , asking about the possible values of dim x for x ∈ I ( F ) n . More precisely, for any n ⩾ 1 , we prove that (*) dim I n = { 2 n + 1 − 2 i | i ∈ [ 1 , n + 1 ] } ∪ ( 2 Z ∩ [ 2 n + 1 , + ∞ ) ) , where dim I n is the set of all dim x for all x ∈ I ( F ) n and all F. Previously available partial informations on dim I n include the classical Arason–Pfister theorem (saying that ( 0 , 2 n ) ∩ dim I n = ∅ ) as well as a recent Vishik's theorem on ( 2 n , 2 n + 2 n − 1 ) ∩ dim I n = ∅ (the case n = 3 is due to Pfister, n = 4 to Hoffmann). The proof of ( * ) 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.