Pfister form

Remarks

The form $\langle\langle a_1, \ldots, a_n \rangle\rangle$ is the diagonal form whose $2^n$ coefficients are the products $\prod_{i \in S} (-a_i)$ over all subsets $S \subseteq \set{1, \ldots, n}$. Example: $\langle\langle a, b \rangle\rangle = \langle 1, -a, -b, ab \rangle$, the norm form of the quaternion algebra $(a, b)_F$. For $a = b = -1$ this is $x_1^2 + x_2^2 + x_3^2 + x_4^2$. Non-example: $\langle 1, 1, 1 \rangle$ is not a Pfister form, since every Pfister form has dimension a power of $2$. A Pfister form is either anisotropic or hyperbolic. Its nonzero values form a subgroup of $F^\times$, and they are exactly its similarity factors. Some authors, such as Lam, write $\langle\langle a \rangle\rangle$ for $\langle 1, a \rangle$ instead of $\langle 1, -a \rangle$.