Editing Clifford algebra (section)

=== Spinor norm ===
{{details|Spinor norm#Galois cohomology and orthogonal groups}}

In arbitrary characteristic, the [[Orthogonal group#The spinor norm|spinor norm]] {{math|''Q''}} is defined on the Lipschitz group by
<math display="block">Q(x) = x^\mathrm{t}x.</math><!-- Note that (−1)^D(x) is the same as \alpha(x), so this expression is the same as \alpha(x)^t x (−1)^D(x). -->
It is a homomorphism from the Lipschitz group to the group {{math|''K''<sup>×</sup>}} of non-zero elements of {{math|''K''}}. It coincides with the quadratic form {{math|''Q''}} of {{math|''V''}} when {{math|''V''}} is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of {{math|−1}}, {{math|2}}, or {{math|−2}} on&nbsp;{{math|Γ<sup>1</sup>}}. The difference is not very important in characteristic other than 2.

The nonzero elements of {{math|''K''}} have spinor norm in the group ({{math|''K''<sup>×</sup>)<sup>2</sup>}} of squares of nonzero elements of the field {{math|''K''}}. So when {{math|''V''}} is finite-dimensional and non-singular we get an induced map from the orthogonal group of {{math|''V''}} to the group {{math|''K''<sup>×</sup>{{px2}}/{{px2}}(''K''<sup>×</sup>)<sup>2</sup>}}, also called the spinor norm. The spinor norm of the reflection about {{math|''r''<sup>⊥</sup>}}, for any vector {{math|''r''}}, has image {{math|''Q''(''r'')}} in {{math|''K''<sup>×</sup>{{px2}}/{{px2}}(''K''<sup>×</sup>)<sup>2</sup>}}, and this property uniquely defines it on the orthogonal group. This gives exact sequences:
<math display="block">\begin{align}
  1 \to \{\pm 1\} \to \operatorname{Pin}_V(K)  &\to \operatorname{O}_V(K)  \to K^\times/\left(K^\times\right)^2, \\
  1 \to \{\pm 1\} \to \operatorname{Spin}_V(K) &\to \operatorname{SO}_V(K) \to K^\times/\left(K^\times\right)^2.
\end{align}</math>

Note that in characteristic {{math|2}} the group {{math|{{mset|±1}}}} has just one element.

From the point of view of [[Galois cohomology]] of [[algebraic group]]s, the spinor norm is a [[connecting homomorphism]] on cohomology. Writing {{math|''μ''<sub>2</sub>}} for the [[Group scheme of roots of unity|algebraic group of square roots of 1]] (over a field of characteristic not {{math|2}} it is roughly the same as a two-element group with trivial Galois action), the short exact sequence
<math display="block"> 1 \to \mu_2 \rightarrow \operatorname{Pin}_V \rightarrow \operatorname{O}_V \rightarrow 1</math>
yields a long exact sequence on cohomology, which begins
<math display="block"> 1 \to H^0(\mu_2; K) \to H^0(\operatorname{Pin}_V; K) \to H^0(\operatorname{O}_V; K) \to H^1(\mu_2; K).</math>

The 0th Galois cohomology group of an algebraic group with coefficients in {{math|''K''}} is just the group of {{math|''K''}}-valued points: {{math|1=''H''<sup>0</sup>(''G''; ''K'') = ''G''(''K'')}}, and {{math|''H''<sup>1</sup>(μ<sub>2</sub>; ''K'') ≅ ''K''<sup>×</sup>{{px2}}/{{px2}}(''K''<sup>×</sup>)<sup>2</sup>}}, which recovers the previous sequence
<math display="block"> 1 \to \{\pm 1\} \to \operatorname{Pin}_V(K) \to \operatorname{O}_V(K)  \to K^\times/\left(K^\times\right)^2,</math>
where the spinor norm is the connecting homomorphism {{math|''H''<sup>0</sup>(O<sub>''V''</sub>; ''K'') → ''H''<sup>1</sup>(μ<sub>2</sub>; ''K'')}}.