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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 I, (25) m=l so that I (m,x) where Z' Ipl i = m, is = a finite ~ > I, i 6 ~ Z' X (p£) l o g l p l , p 6 S ° (k) sum extended over (in p a r t i c u l a r , (26) primes 1 (m,x) subject = 0 to when the condition: 58 follows from lemma ~+iT I m I.

X(P) = g(X) Ipl O, where ~(t,X), p exceptional Proof. for we have (30) Then io~ x + O(x ~) + O(x exp(-c 8 / n ~ x+ ranges over prime zero of L(s,x) ideals of k. )) log(a(x)b(x)) Here ~ in the region defined by denotes (2). T > I, T = {~+itl o=1-@(t), {0~__iT 11+¢(T) where (29) < (1+Itj) n/2 /a(x)b(~x). 5 that z Ip l

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A most easy method for finding many very large prime numbers by Euler L.


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