Pfister form information

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 2ⁿ 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 a₁, ..., a_n of F.^[1] (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.^[2] 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 Iⁿ of the fundamental ideal of the Witt ring of F.^[2]

^ Elman, Karpenko, Merkurjev (2008), section 9.B.
^ ^a ^b Lam (2005) p. 316

[1] Elman, Karpenko, Merkurjev (2008), section 9.B.

[Lam316-2] Lam (2005) p. 316

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