Pfister Form

In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an ''n''-fold Pfister form over ''F'' is a quadratic form of dimension 2^''n'' that can be written as a tensor product of quadratic forms

:$\langle\!\langle a_1, a_2, \ldots , a_n \rangle\!\rangle \cong \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle \otimes \cdots \otimes \langle 1, -a_n \rangle,$

for some nonzero elements ''a''₁, ..., ''a''_''n'' of ''F''. (Some authors omit the signs in this definition; the notation here simplifies the relation to Milnor K-theory, discussed below.) An ''n''-fold Pfister form can also be constructed inductively from an (''n''−1)-fold Pfister form ''q'' and a nonzero element ''a'' of ''F'', as $q \oplus (-a)q$.

So the 1-fold and 2-fold Pfister forms look like:

:$\langle\!\langle a\rangle\!\rangle\cong \langle 1, -a \rangle = x^2 - ay^2$.
:$\langle\!\langle a,b\rangle\!\rangle\cong \langle 1, -a, -b, ab \rangle = x^2 - ay^2 - bz^2 + abw^2.$

For ''n'' ≤ 3, the ''n''-fold Pfister forms are norm forms of composition algebras.Lam (2005) p. 316 In that case, two ''n''-fold Pfister forms are isomorphic if and only if the corresponding composition algebras are isomorphic. In particular, this gives the classification of octonion algebras.

The ''n''-fold Pfister forms additively generate the ''n''-th power ''I'' ^''n'' of the fundamental ideal of the Witt ring of ''F''.Lam (2005) p. 316

Characterizations

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.Lam (2005) p. 324 For anisotropic quadratic forms, Pfister forms are multiplicative, and conversely.Lam (2005) p. 325

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. It was a remarkable discovery by Pfister that ''n''-fold Pfister forms for all ''n'' are multiplicative in the more general sense here, involving rational functions. For example, he deduced that for any field ''F'' and any natural number ''n'', the set of sums of 2^''n'' squares in ''F'' is closed under multiplication, using that
the quadratic form
$x_1^2 +\cdots + x_^2$
is an ''n''-fold Pfister form (namely, $\langle\!\langle -1, \ldots , -1 \rangle\!\rangle$).Lam (2005) p. 319

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 $\langle 1, -1 \rangle$. This property also characterizes Pfister forms, as follows: If ''q'' is an anisotropic quadratic form over a field ''F'', and if ''q'' becomes hyperbolic over every extension field ''E'' such that ''q'' becomes isotropic over ''E'', then ''q'' is isomorphic to ''a''φ for some nonzero ''a'' in ''F'' and some Pfister form φ over ''F''.

Connection with ''K''-theory

Let ''k''_''n''(''F'') be the ''n''-th Milnor ''K''-group modulo 2. There is a homomorphism from ''k''_''n''(''F'') to the quotient ''I''^''n''/''I''^''n''+1 in the Witt ring of ''F'', given by

:$\ \mapsto \langle\!\langle a_1, a_2, \ldots , a_n \rangle\!\rangle ,$

where the image is an ''n''-fold Pfister form. The homomorphism is surjective, since the Pfister forms additively generate ''I''^''n''. One part of the Milnor conjecture, proved by Orlov, Vishik and Voevodsky, states that this homomorphism is in fact an isomorphism . That gives an explicit description of the abelian group ''I''^''n''/''I''^''n''+1 by generators and relations. The other part of the Milnor conjecture, proved by Voevodsky, says that ''k''_''n''(''F'') (and hence ''I''^''n''/''I''^''n''+1) maps isomorphically to the Galois cohomology group ''H''^''n''(''F'', F₂).

Pfister neighbors

A Pfister neighbor is an anisotropic form σ which is isomorphic to a subform of ''a''φ for some nonzero ''a'' in ''F'' and some Pfister form φ with dim φ < 2 dim σ.Elman, Karpenko, Merkurjev (2008), Definition 23.10. The associated Pfister form φ is determined up to isomorphism by σ. Every anisotropic form of dimension 3 is a Pfister neighbor; an anisotropic form of dimension 4 is a Pfister neighbor if and only if its discriminant in ''F''^*/(''F''^*)² is trivial.Lam (2005) p. 341 A field ''F'' has the property that every 5-dimensional anisotropic form over ''F'' is a Pfister neighbor if and only if it is a linked field.Lam (2005) p. 342

Notes

References

*
* , Ch. 10
* {{Citation , title=An exact sequence for ''K''_*^''M''/2 with applications to quadratic forms , author1-first=Dmitri , author1-last=Orlov , author2-first=Alexander , author2-last=Vishik , author3-first=Vladimir , author3-last=Voevodsky , author3-link=Vladimir Voevodsky , journal=Annals of Mathematics , volume=165 , year=2007 , pages=1–13 , doi=10.4007/annals.2007.165.1 , mr=2276765, arxiv=math/0101023

Quadratic forms