ON THE THEORY OF NUMBERS. 147 



the substitution tb>, •=z<a-. Since it can be shown that the numbers conjugate 

 to ¥ (ts-,) are all different from one another, it follows from the definition, 



that the quotient ^ ' represents the number of conjugate ideal prime fac- 

 tors contained in the real prime q, appertaining to the exponent t. If q be a 

 divisor of n, the definition of its ideal factors requires a certain modification, 

 which we cannot here particularize. (See sect. 6 of M. Rummer's Memoir.) 

 The two definitions, corresponding to the cases of q prime to n, and q a 

 divisor of n, enable us. when taken together, to transfer to the general case 

 when n is composite, the elementary theorems already shown to exist when 

 n is prime (see art. 47). We may add that it is easy to prove, in the general 

 as in the special case (see art. 48), that the number of classes of ideal num- 

 bers is finite. 



60. Application to the Theory of the Division of the Circle. — We cannot 

 quit the subject of complex numbers without mentioning certain important 

 investigations in which they have been successfully employed. The first 

 relates to the problem of the division of the circle. In this problem the 



s=p— 2 

 resolvent function of Lagrange 2 6 s x't* (see art. 30) is, as is -well 



5 = 



known, of primary importance. Retaining, with a slight modification, the 

 notation of art. 30, and still representing by A a prime divisor of p — 1, and 



by a a root of the equation =0, let us consider the function F (a, x), 



a — 1 



which is a particular case of the resolvent, and let us represent the quotient 

 F(a, a,-) F(a*, x) , , , . „ . „ , 



F («*+', a) 7 ** ^ e 



lF( a ,x)y=^(ot)4, 2 (a)....+ s - 1 (a)F(a s ,x), ... (1) 

 and in particular, observing that F (a, x) F (a x_1 , x)=p, 



[F (a, x)Y=ph {a) fc (a) . . . . ^- 2 (a), (2) 



a result which is in accordance with the known theorem that [F (a, .r)] A is 

 independent of x and is an integral function of a. only. The resolution of 

 the auxiliary equation of order A, the roots of which are the X periods of 



1 - roots of the equation =0, depends solely on the determination 



of the complex numbers \p x (a), tb (a) i!/a.-2 (a). For when these com- 

 plex numbers are known, we may equate F (a, x) to any Xth root of the ex- 

 pression pyp x (a) \l 2 (a) . . . \p\-2 (a) ; from the value of F (a, x), thus obtained, 

 those of F (a", x), F (a 3 , x) . . . . may be inferred by means of equation (1); 

 and, lastly, from the values of F (1 , a?), F (a, x), . . . F (a* -1 , x), the values of 

 the periods themselves are deducible by the solution of a system of linear 

 equations. To determine the numbers \L 1 (a), \p 2 (a), . . . M. Kummer assigns 

 the ideal prime factors of which they are composed, employing for this pur- 

 pose the results cited in ait. 30. The equation ^- (a) \p/ c (cc~ 1 )=p shows 

 that fa (a) contains precisely \{p — 1) ideal prime divisors of p, and no other 

 complex prime. To distinguish the prime factors of p contained in \pk (a) 

 from those contained in il/jt(a~ ! ) M.Kummer avails himself of the congruence 

 V. of art. 30, viz., 



Let X'= i~ — , and u = y KI , mod p, so that u, u 2 , . . .w* -1 are the roots of 



A 



2l 



