Spinor norm (awaiting review)

The orthogonal group $\GO(n,q)$ is the subgroup of $\GL(n,q)$ fixing a particular quadratic form $Q$, and has the subgroup $\SO(n,q)$ of all matrices with determinant $1$. One can associate to $Q$ a symmetric bilinear form $f \colon V^2 \to \F_q$ (for $V = \F_q^n$) by writing $f(x,y) = Q(x+y) -Q(x) - Q(y)$. Then the reflection in a vector $v \in V$ such that $Q(v) \neq 0$ is the element of $\SO(n,q)$ that acts on $V$ by $$ x \mapsto x - \frac{f(x,v)}{Q(v)} v. $$

We can write any $g \in \SO(n,q)$ as a product of reflections: $g = r_1 \cdot \ldots \cdot r_n$.

The spinor norm is the map $\SO(n,q) \to (\F_q^\times) / (\F_q^\times)^2$ that sends $g$ to the image of $Q(r_1) \cdot \ldots \cdot Q(r_n)$ in the quotient.