Editing Orthogonal group (section)

== Galois cohomology and orthogonal groups ==
In the theory of [[Galois cohomology]] of [[algebraic group]]s, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part ''post hoc'', as far as the discovery of the phenomenon is concerned. The first point is that [[quadratic form]]s over a field can be identified as a Galois {{math|''H''<sup>1</sup>}}, or twisted forms ([[torsor]]s) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the [[determinant]].

The 'spin' name of the spinor norm can be explained by a connection to the [[spin group]] (more accurately a [[pin group]]). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of [[Clifford algebra]]s). The spin covering of the orthogonal group provides a [[short exact sequence]] of [[algebraic group]]s.
: <math> 1 \rightarrow \mu_2 \rightarrow \mathrm{Pin}_V \rightarrow \mathrm{O_V} \rightarrow 1 </math>

Here {{math|''μ''<sub>2</sub>}} is the [[Group scheme of roots of unity|algebraic group of square roots of 1]]; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The [[connecting homomorphism]] from {{math|''H''<sup>0</sup>(O<sub>V</sub>)}}, which is simply the group  {{math|O<sub>V</sub>(''F'')}} of {{math|''F''}}-valued points, to  {{math|''H''<sup>1</sup>(''μ''<sub>2</sub>)}} is essentially the spinor norm, because {{math|''H''<sup>1</sup>(μ<sub>2</sub>)}} is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from {{math|''H''<sup>1</sup>}} of the orthogonal group, to the {{math|''H''<sup>2</sup>}} of the kernel of the spin covering. The cohomology is non-abelian so that this is as far as we can go, at least with the conventional definitions.