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By Dr. Leslie Cohn (auth.)

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Suppose that Z ~ ~ . p ,o,j) B(Z,Hj~)~e ('~) *). X~)e " B ~(X~) -~ --- = . B, Ad n(~)'iHj) = B(Zn'l, AC n(~)'IHj-Hj) = B(Z ,Hik(~')) - B(Z ,HiN) " - B(Z,H j ~ ). 3. li. is such that e x(H(~) ) is a polynomial function; and suppose that e ~(H(~)) = ~ji(B)bi i=l (bill) 35 is the factorization of e k'H'-''( ( ~ into irreducible polynomial functions. ,a). ;a and Z ~ ~ , Remark. (Z e ~ , ~ £ ~). Ji divides q(Z)J i. Let Ji(~) and Cj. be as in the preceeding corollary. (VI~) = 0 for V E ~ M ' ~ e N. (~) = I when ~ = e.

First suppose that b = Z a ~ . 2 givewthe statement. Let ~(J) = ~ + ~ c + "'" + ~ c j (J -• 0). Assume that the proposition is valid if b s ~(J) and suppose that b' £ 9~(J+l). Clearly, it suffices to assume that b' has the form b' = Zb with Z ¢ ~ , b e ~(J). Let d(b) denote the reduced degree of Fj(~I~)(b) as a polynomial function on N; and let Cj(Zb) = B''(c(b) + d(J)). ,£), T(Zb) fN-J(~)eiv-0(H(~) )~(~m)dn £ = ~j=l~i~+P,~>fNJ(~)ev(n)B(Z, Hj~)(~T(I @ Fj(~l~)(b))$)(~Im)d~ + f~(q(Z)J)(~)ev(~)(~x(l @ Fj(~l~)(b))~)(~Im)d~ + /~J(~)ev(~)(~ (i @ q(Z)Fj(vl~)(b))~(~Im)d~ - ~N-J(~)ev(~)~B(Z, vjn)(IT(l ® Fj(~l~)(b)Vj)$)(~Im)d~ = ~J(~)ev(~)(Ix(l ® {Fj(~l~)(b)Fj(vl~)(Z) + q(Z)Fj(vl~)(b)})~)(~Im)d~ = /~j(~)ei~-P(H(~))(~ (i ® Fj(vl~)(Zb))~)(~Im)dH.

Hence, 45 q(XI)F(2)(~I~)(X2) - ~(X2)F(2)(~I~)(XI) + [F(2)(~I~)(XI) , F(2)(vI~)(X2 )] = ~([~2(XI{-I), ~l(Xl{'l)] . ~I(XI[-I)] + [~(XI~'I), X2~-I)] ) = ~([XI~-I,x2~-I]) = F(2)(vI~)([XI,X2]), as claimed. b. If XI, X2 e ~ l , then q(Xl)¢i(x2) - Q(x2)~i(xi) = ~/[Xl,X2]). Proof. We have q(X1)q(X2)I = q ( X 1 ) ( ¢ i ( X 2 ) I ) = (q(X1)¢I(X2))I + ¢I(X2)q(XI)I = (q(X1)¢l(X2))l + ¢I(XI)¢I(~)I, Hence, [q(Xl) , q(X2)]I = {q(Zl)¢i(X2) - q(X~(Xl)}i" Also, [q(Xl) , q(X2)]I = q([XI,X2])I = ¢I([XI,X2])I. Therefore, ¢i([Xl,X2])I = {~(×l)¢i(x 2) - q(z2)¢i(Xl)}I.

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