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2 Clifford Algebras 17 √ is known that C ∼ C + over K( (q)) where the meaning of ∼ √ is that all simple (q)). 3 Theorem • If dimK E is even, the Clifford algebra C(E, q) is a central simple algebra over K. The principal automorphism π is then an inner automorphism of C. 2 = • If dimK E is odd and if eN (q) ∈ / (K × )2 , then C(E, q) is a central simple √ ˜ algebra over K = K( (q)) = {λ1C + µeN /λ, µ ∈ K}, a quadratic extension of K, and C = C + ⊕eN C + is an extension of C + . Finally, π(λ1C +µeN ) = λ1C −µeN is the unique automorphism of K˜ different from the neutral element, which leaves K ˜ invariant in the Galois group of K: π(a+ + eN b+ ) = a+ − eN b+ with a+ , b+ ∈ C + .

2 Definitions ˜ is the multiplicative group of invertible elements g in The Clifford regular group G C(E, q) that satisfy, for any x ∈ E, π(g)xg −1 = y ∈ E. ˜ belongs The linear transformation ψ(g) : x → π(g)xg −1 induced by g ∈ G to the orthogonal group O(q) of E relative to q. The mapping g → ψ(g) is a ˜ into O(q). Therefore, ψ is a representation of G ˜ on E, called homomorphism from G ∗ ˜ ˜ the regular vector representationof G. The kernel of ψ is K . G is identical to the subset of C(E, q) consisting of products of regular (or nonisotropic) vectors of E, ˜ can be, equivalently, defined as the multiplicative group formed, with the unit and G element 1C (E, q), by products of regular vectors of E.

J. P. Serre, Applications algébriques de la cohomologie des groupes, op. , p. 603. 1. 39 n Cn,0 CnC = Cn,0 ⊗ C C0,n 1 R⊕R C C⊕C 2 m(2, R) H m(2, C) 3 m(2, C) H⊕H m(2, C) ⊕ m(2, C) 4 m(2, H) m(2, H) m(4, C) 5 m(2, H) ⊕ m(2, H) m(4, C) m(4, C) ⊕ m(4, C) 6 m(4, H) m(8, R) m(8, C) 7 m(8, C) m(8, R) ⊕ m(8, R) m(8, C) ⊕ m(8, C) 8 m(16, R) m(16, R) m(16, C) C0,n ⊗ C Thus, for example, C14,0 m(64, H) since 14 ≡ 6 (mod 8) and C6,0 Since Cr,s ⊗ C1,1 Cr+1,s+1 , if we assume that r > s, we obtain Cr−s,0 ⊗ C1,1 ⊗ · · · ⊗ C1,1 , s factors 38 M.