122 REPORT 1860. 



41. Elementary Definitions relating to Complex Numbers. — Let X be a prime 



a x — 1 

 number, and a a root of the equation r- = 0; then any expression of the 



form 



F(a)=a + a 1 a + a 2 a 2 + -\-a K _ 3 a x ~ i .... (A.) 



in which a , a v a 2 a *.-» denote real integers, is called a complex inte- 

 gral number. To this form every rational and integral function of a can 

 always be reduced ; and it follows, from the irreducibility of the equation 



a x — 1 



t-=0, that the same complex number cannot be expressed in this 



a. — 1 



reduced form in two different ways. The norm of F(a) is the real integer 



obtained by forming the product of all the X — 1 values of F(a), so that 



N . F(«) = N . F(<r)= . . . =N . F(a»- 1 )=F(a) . F(«") . F(a a ) . . . T(**- 1 ). 



The operations of addition, subtraction and multiplication present no pecu- 

 liarity in the case of these complex numbers ; by the introduction of the 

 norm, the division of one complex number by another is reduced to the case 

 in which the divisor is a real integer. Thus 



f(a) _ /Q)F(y)F(y ) .... F(«*-Q . 

 F(«) N.F(a) 



and /(a.) is said to be divisible by F(a) when every coefficient in the pro- 

 duct /(a)F(a 2 )F(a 3 ) ... F(a x_1 ), developed and reduced to the form (A), 

 is divisible by N.F(a). When /(a) is not divisible by F(a), it is not, in 

 general, possible to render the norm of the remainder less than the norm of 

 the divisor; and it is owing to this circumstance that the common rule for 

 finding the greatest common divisor is not generally applicable to complex 

 numbers. If, in the expression (A), we consider the numbers a , o 1 ...ax_2 

 as indeterminate*, the norm is a certain homogeneous function of order X — 1, 

 and of X — 1 indeterminates ; so that the inquiry whether a given real number 

 is or is not resoluble into the product of X — 1 conjugate complex factors, is 

 identical with the inquiry whether it is or is not capable of representation by 

 a certain homogeneous form, which is, in fact, the resultant of the two forms 



rt ( /p x - 2 +c 1 a? x - 3 y H |-«a-22/ x-2 > 



and x x - 1 +x*-- 2 i/ + x x - 3 y 2 -\- +y>-- 1 . 



The problem is considered in the former aspect by M. Kummer, in the latter 

 by Dirichlet. The methods of Dirichlet appear to have been of extreme 

 generality, and are as applicable to complex numbers, composed with the 

 powers of a root of any irreducible equation having integral coefficients, as 

 to the complex numbers which we have to consider here. Nevertheless, in 

 the outline of this theory which we propose to give, we prefer to follow the 

 course taken by M. Kummer: for Dirichlet' s results have been indicated 

 by him, for the most part, only in a very summary manner *; nor is it in any 

 case difficult to assign to them their proper place in M. Kummer's theory ; 

 while, on the other hand, it would, perhaps, be impossible to express ade- 

 quately, in any other form than that which M. Kummer has adopted, the 

 numerous and important results (including the law of reciprocity itself) con- 



* See his notes in the Monatsberichte of the Berlin Academy for 1841, Oct. 11, p. 280; 

 1842, April 14, p. 93; and 1846, March 30; also a letter to M. Liouville, in Liouville's 

 Journal, vol. v. p. 72 ; a note in the Comptes Rendus of the Paris Academy for 1840, 

 vol. x. p. 286 ; and another in the Monatsberichte for 1847, April 15, p. 139. 



