By Baruch Z. Moroz

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Suppose KIK' through shape Proof. Lemma p such that I. then is of Galois extension G(Klk). Corollary type, K'Ik. G(k'Ik) 2 suffices 5. irreducible. Let Then in p is of the k'Ik. the argument in used in the proof of with if p is not R(k). and suppose P = Pl ~ p 2 One defines W(k) is said to be primitive, p 6 R(k) Ro(Klk) factors this statement. 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.

6 R(k), L(s,xj) X = tr ~ / O for and suppose I Re s > ~ , that L(s,x) I < j < m.

33) T and numerical. N E ~. 40 Theorem 1. Suppose that p is of A W B(X) := 12n6g(x) (a(x)+l) The f o l l o w i n g estimate type and let X = tr ยข, lap~tpl [~ (l+ItpJ) ~~ (1+ P6S1 P6S2 2 (34) ) 2 holds: 2 A(x,x) = xP(x,log x) + 0 (B(x)xI-~ + C(log x) nd), (AW 35) C m where N:= g(x) >__ I ! :~], 3 P(X,t) implied by O e - s y m b o l Proof. By P(X,t) = 0 is a p o l y n o m i a l when constant g(X) = O, may d e p e n d on of degree g(x)-I x > 2; ~ > O. e and Here when the nd(x). (3), oo L(S,X) with an e f f e c t i v e l y Therefore p the line This Z n=1 nen-Sc1 (e,nd(x)), computable (in terms of e > 0 (36) nd(x) and c) C I > O.