For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product of quadratic forms for some nonzero elements a1, ..., an of F.[1] (Some authors omit the signs in this definition; the notation here simplifies the relation to Milnor K-theory, discussed below.)
The n-fold Pfister forms additively generate the n-th power I n of the fundamental ideal of the Witt ring of F.[2] A quadratic form q over a field F is multiplicative if, for vectors of indeterminates x and y, we can write q(x).q(y) = q(z) for some vector z of rational functions in the x and y over F. Isotropic quadratic forms are multiplicative.
[4] For n-fold Pfister forms with n ≤ 3, this had been known since the 19th century; in that case z can be taken to be bilinear in x and y, by the properties of composition algebras.
For example, he deduced that for any field F and any natural number n, the set of sums of 2n squares in F is closed under multiplication, using that the quadratic form
[5] Another striking feature of Pfister forms is that every isotropic Pfister form is in fact hyperbolic, that is, isomorphic to a direct sum of copies of the hyperbolic plane
There is a homomorphism from kn(F) to the quotient In/In+1 in the Witt ring of F, given by where the image is an n-fold Pfister form.
One part of the Milnor conjecture, proved by Orlov, Vishik and Voevodsky, states that this homomorphism is in fact an isomorphism kn(F) ≅ In/In+1.
The other part of the Milnor conjecture, proved by Voevodsky, says that kn(F) (and hence In/In+1) maps isomorphically to the Galois cohomology group Hn(F, F2).
A Pfister neighbor is an anisotropic form σ which is isomorphic to a subform of aφ for some nonzero a in F and some Pfister form φ with dim φ < 2 dim σ.