By Scott R.F.

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R(k) [87], p. 10, and that X = tr p. therefore extension ~ W(k)/W(k'), to establish p for p 6 R (Klk). o induced by any other r e p r e s e n t a t i o n We remark p E R(K'Ik) Then there is a finite abelian exten- for a finite Galois Since See type and that NK/K, (C K) ~ Ker p ; Thus A representation Proof. for a finite Every closed subgroup of finite index in W(k') Proposition p E Ro(KIk) Klk. that a finite Galois sion = {p 6 R(KIk)IC K _~ Ker p} that p Pl E gr(k) is primitive and and P2 6 Ro(k).

P - 3 Y: = I m p. ,~ 1 O (A(x, t) ) (23) It-~1 >_I i~-tl 2 12+it-pl Since > I for each Q, estimate (20) (23). Let X E gr(k) and suppose that L(s,X) ~ O for Then Z X(P) Ipl

X(P) = g(X) Ipl

### A treatise on the theory of determinants by Scott R.F.

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