79 



or 



fsi fisi — i; 



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

 If we let B = P^= we get from (3) 



fSl f(8,— ll 



F(«, pW) = F(/". pT-)-'Fia'' , p!=) 



and hence by (2) 



(4) F(«, P'T^')=0, modP^ 

 By a similar reasoning we also get, 



(5) F ( «, Pf Pf ) = Omo d P^ and hence by (4) and (5) . 



(6) F(«, P^ P^) ^OmodP-^=P^. 



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

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



fs f (S — 1 I 



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



(7) F(«, APO^OmodA. 



Now let A = CQ' where Q is a prime ideal and C prime to Q. Tlien, 

 ft f'(t-^ii 



F(«, AP'^) = F(« '' ,CPn — F('/' , CP'^) where q is the ra- 

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

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



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



(9) F(«, AP'')=0 mod AP\ 



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+l distinct prime 

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

 for any A. 



On the Class Number of the Cyclotomic Xumberfield 



Jacob Westluxd. 



[By title.] 



[Will appear in Transactions Amcripan Mathematical Society. Vol. IV: 2.] 



