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I. §2. CATEGORIES SATISFYING KRULL-SCHMIDT THEOREMS 19 IAI=c d im A for a fixed c>I. In particular, this is the case if F is an algebraically closed field of characteristic zero. In that case, with a finite number of exceptions, the simple finite-dimensional Lie algebras are represented by four infinite classes of algebras. In common notation, these are listed in the manner: (i) An (n5'= I), dim A n=n 2+2n; (ii) s, (n5'=2), dim Bn=2n 2+n; (iii) C; (n5'=3), dim Cn=2n 2+n; (iv) D n (n5'=4), dim D n=2n 2-n.

Therefore the corresponding polynomial i, has degree at most k + I in t, and a calculation shows that gk = I -kt+[2k+Hk+ l)(k-2)]t 2+ .... The coefficient of t 2 is greater than (k! I) when k > 1. 6. Thus 13 does not possess a finite '-formula when k > 1. k We conclude this section with a more general proposition, whose statement is illustrated by the above examples. 10. Proposition. t", r>1 where ar=a(r) is a polynomial of degree s in I' whose coefficients are rational integers. If a(r) is not identically I, -I or 0, then at most a finite number of the point-wise powers possess finite '-formulae.

We note that the first seven formulae involve PIM-functions. Therefore it is possible to make use of the monomorphism ": PIM --C [[t]J defined in § 4, in the following way: Suppose that f is a PIM-function and that it has already been proved that J is a finite product of the form J = II (1 i t'"I)-k" where the m, and k, are integers and m, >0. Then note that, since the substitution operation z-s mz defines a continuous algebra endomorphism of 42 ARITHMETICAL FUNCTIONS CH. 2. §6. Dir (G), say. Therefore and it follows that Hence j(z) = II rCdmiz)]k,.

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Abstract analytic number theory. V12 by Knopfmacher

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