# 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. 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.

## 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. For anisotropic quadratic forms, Pfister forms are multiplicative, and conversely.
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 $x_{1}^{2}+\cdots +x_{2^{n}}^{2}$ is an n-fold Pfister form (namely, $\langle \!\langle -1,\ldots ,-1\rangle \!\rangle$).
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 aφ 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
    $\{a_{1},\ldots ,a_{n}\}\mapsto \langle \!\langle a_{1},a_{2},\ldots ,a_{n}\rangle \!\rangle ,$
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 k$_n$(F) ≅ I$^n$/I$^{n+1}$. 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, F$_2$).

## Pfister neighbors

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. 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. 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.