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Of L£(K) Si> in l. 9 follows. 10: L£,tor(K o rr. ) Now we consider the general case. 11 is exact. 13 follows. 13 « s >1 < t > -2 < 1 »p s IITI , >1 < t » > 1 < t > )p Thus contributes at most a be divisible by all primes dividing Pjl'ITl ,~(s)~(t) At 1. 15: E is actually 0 D. 1 in e'K For example, this happens when K Nq:;(E) = -1 with In the type I case either a real subfield of a cyclotomic field. Di is a Quaternion algebra or In the latter case, it is convenient to reverse the viewpoint taken for the type II algebras above, and instead measure the deviation between the image of the and the entire image.

2 : ~-symmetric If -1 )= AT(A) -1 2 -1 T (M)T(A)A Given any involution of matrix a = TA , A so that and a A is +1 with M (D) n ~ 2 M F a M. tive or negative type respectively. We say that two involutions there is an automorphism means that (M (D), T) n a and of T and of so that Mn(D) Mn(D) T • a are equivalent if a 0 a. 3: l i A' = BAr (B) ,where then TA Proof: is equivalent to Define TA 0 a(M) -1 = is a non-singular matrix over D, T , . A B MB a(M) B AT(B -1 Then we have -1 MB)A 1 1 T on Mn(D) in CD up to equivalence AT(B)T(M)T(B- )A- = B-1A'T(M)A,-lB Remark: FT Thus, the classification of involutions equal to a given subfield of index 1 or 2 with is equivalent to the classification of non-singular E-a-symmetric matrices A under the equivalence relation 0' some non-singular matrix a is a particular involution with Fa F.

10: L£,tor(K o rr. ) Now we consider the general case. 11 is exact. 13 follows. 13 « s >1 < t > -2 < 1 »p s IITI , >1 < t » > 1 < t > )p Thus contributes at most a be divisible by all primes dividing Pjl'ITl ,~(s)~(t) At 1. 15: E is actually 0 D. 1 in e'K For example, this happens when K Nq:;(E) = -1 with In the type I case either a real subfield of a cyclotomic field. Di is a Quaternion algebra or In the latter case, it is convenient to reverse the viewpoint taken for the type II algebras above, and instead measure the deviation between the image of the and the entire image.