the one here by a factor of −1, 2, or −2 on Γ1. The difference is not very important in characteristic other than 2. The nonzero elements of K have spinor norm in the group (K×)2 of squares of nonzero elements of the field K. So when V is finite-dimensional and non-singular we get an induced map from the orthogonal group of V to the group K×/(K×)2, also called the spinor norm. The spinor norm of the reflection about r⊥, for any vector r, has image Q(r) in K×/(K×)2, and this property uniquely defines it on the orthogonal group. This gives exact sequences: Note that in characteristic 2 the group {±1} has just one element. From the point of view of Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the 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 short exact sequence yields a long exact sequence on cohomology, which begins The 0th Galois cohomology group of an algebraic group with coefficients in K is just the group of K-valued points: , and , which recovers the previous sequence where the spinor norm is the connecting homomorphism . Spin and Pin groups In this section we assume that is finite-dimensional and its bilinear form is non-singular. The pin group is the subgroup of the Lipschitz group of elements of spinor norm , and similarly the spin group is the subgroup of elements of Dickson invariant in . When the characteristic is not , these are the elements of determinant . The spin group usually has index in the pin group. Recall from the previous section that there is a homomorphism from the Lipschitz group onto the orthogonal group. We define the special orthogonal group to be the image of . If does not have characteristic this is just the group of elements of the orthogonal group of determinant . If does have characteristic , then all elements of the orthogonal group have determinant , and the special orthogonal group is the set of elements of Dickson invariant . There is a homomorphism from the pin group to the orthogonal group. The image consists of the elements of spinor norm . The kernel consists of the elements and , and has order unless