Embeddability of quadratic forms in Pfister forms, by Detlev W. Hoffmann and Oleg T. Izhboldin

Let F be a field of characteristic different from 2 and let q be an anisotropic quadratic form over F. The form q is called m-embeddable if q is similar to a subform of an anisotropic m-fold Pfister form. This property can be expressed in terms of Milnor k-theory. By a theorem of Elman-Lam, the form q is m-embeddable if and only if the kernel of the restriction map of the graded ring of Milnor K-theory mod 2 of F to that of F(q) contains a nontrivial symbol of degree m. Let m(q) (resp. m_{ext}(q); resp. m_{ptr}(q) be the smallest integer m such that q is m-embeddable over F (resp. over an extension of F; resp. over a purely transcendental extension of F). We study the possible values of the invariants m(q) and m_{ext}(q) for forms of a given dimension d. We also prove that the invariant m_{ptr} depends only on m(q) and m_{ext}(q), more precisely that it is the minimum of m(q) and m_{ext}(q)+1. In particular, this implies that any form of dimension less or equal to 2^n+1 is (n+2)-embeddable over a suitable purely transcendental extension of the field F. As an application, we show that for certain generic quadratic forms q of dimension d, 2^{n-1} < d < 2^n+1, each system of homogeneous elements which generate the kernel of the above mentioned restriction map will necessarily contain elements of degree d-1 and of degree less that n+2.

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