152 REPORT — 1860. 



Secondly, let one of the numbers x, y, z (for example, z) be divisible by 

 1 — a; it will be convenient to consider the equation in the generalized form 



a>+/ = E(»(l-a)'"'V, (1) 



in which x, y, and z are all prime to 1 — a, and E(a) is any unit. We may 

 assume that the values of x and y are of the form 



x=a + (l — a.) 2 X, 

 y=b + (l-ay-Y, 



a and b being prime to X, but satisfying the relation a+£=0, mod X. 

 In the first place, mmust be greater than 1. For since x x =a\ &n(\y K = b*, 

 mod (1 — a)* +1 , if x K +y K be divisible by (I — a)\ a K +b K is divisible by X ! , and 

 therefore x K +y K by (1 — a)* +1 . Again, each of the factors x+ay, x+a?y, 

 . .. x-\-a x ~ l y is divisible once, and once only, by 1 — a; whence it follows 

 that x+y is divisible by (I— a) mK ~ K+1 , and that no two of the X factors of 

 x K +y K have any other common divisor than I— ex.. Hence the X factors 

 x+y %+a.y x+a K ~ 1 y 



(1— «)«*-*+!' T=o"' 1-a 



are relatively prime, and may be represented by expressions of the form 



e o(«)0o A > «i(*)0i\ e*_,(a)^_,\ 



e o (°0> e i (*)> • • • representing units, and f*, ff, Xth powers prime to 



1 — a. Eliminating x and y from the three equations 



x+y =«.(«) (1-ar^+V. 



x + a r y=e r (cc)(l— a)^./, 

 x + a, s y=e s (a.)(l — a)0 s \ 

 we obtain a result of the form 



r x + e < a )^ x =E I ( a )(l-a)^- , ' x ^, ... (2) 



e(a) and E, (a) denoting two units. But, as in the former case, it may 

 be shown that r A and f s K are congruous, mod X, to real integers, and 

 (1 — a) (m_1 *=(), mod X, because m>l. Hence e (a) is also congruous to 

 a real integer for the modulus X, and is therefore a perfect Xth power by a 

 property of every non-exceptional prime (see art. 52). The equation (2) 

 therefore assumes the form 



* 1 ,l +^ A =E 1 ( a )^(l-«) {B! - ,)A . 



If, therefore, the proposed equation (1) be possible, it will follow, by suc- 

 cessive applications of this reduction, that the equation 



x x +y K =E(a)(l-afz K 



is also possible. But this equation has been shown to be impossible; the 

 equation (1) is therefore p.Iso impossible. 



62. Application to the Theory of Numerical Equations. — In the Monats- 

 berichte for June 20, 1853 (see also the Monatsberichte for 1856, p. 203), 

 M. Kronecker has enunciated ihe following theorem: — 



" The roots of any Abelian equation, the coefficients of which are integral 

 numbers, are rational functions of roots of unity." The demonstration of 

 this theorem (Monatsberichte for 1853, p. 371-373) depends on a compa- 



