# Spinor norm

## Definition

Suppose $k$ is a field, $V$ is a vector space over $k$, and $b$ is a nondegenerate symmetric bilinear form. Let $G$ be the orthogonal group corresponding to $b$. The spinor norm is a homomorphism from $G$ to the multiplicative group modulo squares of $k$:

$\! G \to k^*/(k^*)^2$

i.e., it is a homomorphism that sends any element in the orthogonal group to an element in the multiplicative group of $k$ modulo the squares in that group.

The homomorphism is defined as follows: any element of $G$ arising as reflection orthogonal to a vector $v$ is sent to the value $b(v,v)$ modulo $(k^*)^2$. This extends to a well defined and unique homomorphism on all of $G$. For characteristic not equal to 2, the reflection orthogonal to $v$ is defined as:

$\! x \mapsto x - 2v\frac{b(v,x)}{b(v,v)}$

For characteristic equal to 2, the reflection orthogonal to $v$ is defined as:

$\! x \mapsto x - v\frac{b(v,x)}{b(v,v)}$

Also note that for such a reflection map to exist, $b(v,v)$ must be nonzero, so the map does indeed go to $(k^*)/(k^*)^2$.

Note that different choices of $v$ that are scalar multiples of each other define the same reflection map. That is why the spinor norm is defined only as a map to $k^*/(k^*)^2$ and not as a map to $k^*$.