Pfister form

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In mathematics, a Pfister form is a particular kind of quadratic form over a field F (whose characteristic is usually assumed to be not 2), introduced by A. Pfister in 1965. A Pfister form is in 2n variables, for some natural number n (also called an n-Pfister form), and may be written as a tensor product of quadratic forms as:

\langle \langle a_1, a_2, ... , a_n \rangle \rangle \cong \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle \otimes ... \otimes \langle 1, -a_n \rangle,

For ai elements of the field F. An n-Pfister form may also be constructed inductively from an n-1-Pfister form q and an a in F, as q \oplus (-a)q.

So all 1-Pfister forms and 2-Pfister forms look like:

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

n-Pfister forms for n ≤ 4 are norm forms of composition algebras. In fact, in this case, two n-Pfister forms are isometric if and only if the corresponding composition algebras are isomorphic.

[edit] Reference

  • T. Y. Lam, The Algebraic Theory of Quadratic Forms, Ch. 10
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