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G by N(o) = &(o)c 2 (o) · · · ct- 1 (0). ,(o)} with i(G(F)). ) is a union of conjugacy class in G(F). If wi, w 2 E G(F) and i( w 1 ) is conjugate to i( w 2 ) in G(F), then w 1 is conjugate to w2 in G(F) and hence (j is a conjugacy class. If = (x 1 ,x 2 , ... ,xt), then o N(o) = (1,x;- 11x1, ... ,(x1xz, ... ,Xt-1)- 17(x1x2, ... ,Xt-1)) where/= x 1 x2 ... Xf, and N(fJ) is conjugate in G(F) to i('Y). )(7).

If T = T(L" ,o:"), where L" is not a field, then Tis isomorphic to (Gm)K/E for some extension K/E. In this case, 26 Chapter 3 H 1 (F, T) = H 1 (E, (Gm)K/E) by Shapiro's lemma, and H 1 (E, (Gm)K/E) = 0 by Theorem 90. It is clear that NL' IL is surjective in this case, and (a) follows. 2), the triviality of A(T) follows by Theorem 90 in the local case and by the vanishing of H 1 for the idele class group in the global case. If T = T(L',a'), where L' is a field, then, then (a) follows from theorem 90 and the cohomology sequence associated to the exact sequence defined by the norm: 1 --t T --t (Gm)L' /F --t (Gm)L/F --t 1.

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Cohomology theory by S. T Hu


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