Pfister form

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 a_i 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:

$\langle\langle a\rangle\rangle\cong \langle 1, a \rangle \cong x^2 %2B ay^2$ .

$\langle\langle a,b\rangle\rangle\cong \langle 1, a, b, ab \rangle \cong x^2 %2B ay^2 %2Bbz^2 %2Babw^2.$

n-Pfister forms for n ≤ 3 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.

References

Lam, Tsit-Yuen (2004), Introduction to quadratic forms over fields, Graduate Studies in Mathematics, 67, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1095-8, MR 2104929 , Ch. 10