Classes of Congruent Integers. £} 



m=p"q\"j where p, q, ... are distinct prime ideals, and if 9)t, Sß, Q, ... 

 be the groups corresponding to the moduli nt, p', q*, ... respec- 

 tively, then it can be shewn that the invariants of 9)ï are no more 

 than those of ?ß, D, ... taken together. Hence hereafter we shall 

 always suppose that m=p", p being a prime ideal, and n a positive 

 integer. 



The norm of p is A T (p)=]/, p being a natural prime divisible 

 by p, and /the degree of p. Then the order of group 3)1 is given 

 by 



? (p")=Mp") (i- ^)=P" a - 1) (2^-D 



Now, since p f( "~ IJ and j^—l are relatively prime to each other, 

 there are in 3)1 p f( " _I) elements whose orders are divisors of p f( * -1) , 

 forming a subgroup of 5DÎ, say 2Ï; and also jp-* — 1 elements whose 

 orders are divisors of p f — 1, forming another subgroup 33; and 

 2)i=2(93. Thus Ave need confine our investigation to 3t and 33. 



As for the subgroup S3, let w be a primitive root of p, i.e. a 

 number which belongs to the exponent <p(y)=p f —l with respect to 

 mod. p. Then the exponent to which o> belongs with respect to 

 mod. p" is necessarily divisible by p f —l. Let this exponent be 

 (p f — l)e; then the p f — 1 numbers, 1, <o\ «?",... w^~ 2 ' e , are all in- 

 congruent with respect to mod. p", and evidently all belong to the 

 exponents which are divisors of p f — 1. Hence the p f — 1 classes 

 represented by these numbers are the totality of the elements of 

 33. Thus we obtain the following result : 



33 is a cyclic group, which can he represented as the poivers of an 

 clement (e.g. the class represented by o/) ivhose order is p f —l. If 

 2> f —l=p" 1 p"' 2 ...pT, where p u p. 2 ,...p /l are distinct prime factors, then the 

 invariants of 33 are p" 1 , pT-, ...pf/ 1 . 



Next, let us investigate the other subgroup St. Let p'' be the 

 highest power of p contained in p. For convenience we dis- 

 tinguish the following four cases : 



1) Cf. Dirichlet: Zahlentheorie, §. 131, §. 180, IT, or Weber: Algebra, Bd. II, §. 18 and 

 §. 166 [5]. 



