### 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.

Detlev W. Hoffmann <detlev@math.univ-fcomte.fr>

Oleg T. Izhboldin <oleg@mathematik.uni-bielefeld.de>