ON THE THEORY OP NUMBERS. 145 



strations of his law of reciprocity, which, though they also depend on the 

 consideration of complex numbers containing w, yet do not require the same 

 complicated preliminary considerations. 



59. Complex Numbers composed of Roots of Unity, of which the Index is 

 not a Prime. — In a special memoir (see the list in art. 41, note, No. 16), 

 M. Kummer has considered the theory of complex numbers composed with 

 a root of the equation w"=l, in which n denotes a composite number. The 

 primitive roots of this equation are the roots of an irreducible equation of the 

 form 



n 



n(wP— i)n(w?i^/>3— i) 



PvPvP* '"• denoting the different prime divisors of n*. If ^ (n) be the 

 number of numbers less than n and prime to it, F (w) is of the order \p («), 

 and every complex number containing w can be reduced (and that in one way 



only) to the form / (w)=a + « 1 w + a 2 w a + +«j w _iw , f ( " ) - 1 . The 



numbers conjugate to/(w) are the \p (?i) numbers obtained by writing in 

 succession for w the y (?i) primitive roots of w n =l ; and the norm off (<o) 

 is the real and positive integer produced by multiplying together the \p (n) 

 conjugates. If q be a prime number not dividing n, the sum 



in which the series of terms is to be continued until it begins to repeat itself, 

 is termed a period. The n periods m^ vr 2 , . . . nr rt remain unchanged if for w 

 we write w?, <</'% etc. Hence, if q appertain to the exponent t for the modu- 

 lus n (i. e. if q satisfy the congruence q l = 1, mod n, but no congruence of a 

 lower order and similar form), the number of different numbers conjugate to 



a given complex number containing the periods only is at most zS™). For 



brevity, a complex number containing the periods only — for example, the 

 number 



C + C 1 VI l + C 2 TB. : ,-\- .. . . + C n lZ m 



may be symbolized by/(ra - 1 ), so that 



/Oa) = c 0+ c i «*+« a W lt + .... +Cn Vnk. 



If 1, r v r 2 ,.,. are a set of- ^ numbers prime to n and such that the quo- 

 tient of no two of them (considered as a congruential fractionf) is congruous 

 for the modulus n to any power of q, the numbers conjugate to/ (or) may be 



* The irreducibilit y of the equation . = when n is a prime was first established by 



x—\ 

 Gauss (Disq. Arith. art. 341). For other and simpler demonstrations of the same theorem, 

 see the memoirs of MM. Kronecker (Crelle, xxix. p. 280, and Liouville, 2nd series, vol. i. 

 p. 399), Schoenemaun (Crelle, vol. xxxi. p. 323, vol. xxxii. p. 100, & vol. xl. p. 188), Eisensteiu 

 (Crelle, vol.xxxix. p. 166), and Serret(Liouville,vol.xv.p.296). The principles on which these 



m 



demonstrations depend suffice to establish the irreducibility of the equation— — — = °> 



x^ —1 

 but they fail, as M. Kronecker has observed, to furnish the corresponding demonstration 

 when n, as in the text, is a product of powers of different primes. This demonstration was 

 first given by M. Kronecker (Liouville, vol. xiw p. 177), who has been followed by M. De- 

 dekmd (Crelle, vol. liv. p. 27), and by M. Arndt (ib. lid. p. 178). 



t For the definition of a congruential fraction see art. 1 1. 

 1860. 



