f(si-i; 



(3) F(«, BP^)^F(«''\ B) — F(«'' , B). 



If we let BzziPj' we get from (3) 



fiBi— ll 



FC", p'=p'')=:F(« , p'^)-F(« , p'=) 



and hence by (2) 



(4) F(«, P^P^)=eO, modP^ 

 By a similar reasoning we also get, 



(5) F(o, P'r p! )EEOmodP^and hence by (4) and (5). 



(6) F(", P^ P^) ^OmodP^^P^. 



We now assume that for an arbitrary a the function F (", A) is divis- 

 ible by A, then if P be any prime ideal not contained in A we have by (3) 



ts t(3 — 1) 



F ( «, A P-^) = F ( «'' , A ) — F ( "'' , A) and hence, 



(7) F(«, AP-^)— OmodA. 



Now let A^CQf where Q is a prime ideal and C prime to Q. Tlien, 

 It t'(t-i) 



F{n, APO = F(« '^ ,CP^)— F(« , GP^) wliere q is the ra- 

 tional prime divisible by Q and t the degree of Q, and since by our assump- 

 tion tlie two terms on the right side are divisible by CP" it follows tliat, 



(H) F(", AP-) ^0 mod CP% and hence, 



(9) F(«, AP^)=OmodAP^ 



Hence if F( ", A) is divisible by A when A contains n distinct prime 

 factors it is also divisible by A when A contains n-fl distinct prime 

 factors. Making use of (4) we then find that F( «, A) is divisible by A 

 for any A. 



On tee Class Number of the Cyclotomic Xumberfield 



2 7r/ ! 



K 



e 



p" J 



Jacob Westluxd. 



[By title.] 



[Will apiiear in Transactions American Mathenintical Society, Vol. lY: 2.] 



