132 REPORT — 1860. 



The prime number \ is the only real prime excluded from the preceding 

 considerations. Since X=(l— a)(l— a 2 ) .. ..(1— aX- 1 ), it appears that the 

 norm of 1— a is a real prime, and therefore 1— a cannot be resolved into 

 the product of two factors, except one of them be a unit. Again, because 

 the necessary and sufficient condition for the divisibility of a complex number 

 by i_ H } 3 that the sum of the coefficients of the complex number should 

 be congruous to zero for the modulus X, and because the sum of the coeffi- 

 cients of a product of complex numbers is congruous, for the modulus X, to 

 the product of the sums of the coefficients of the factors, it appears that if 

 the norm of a complex number is divisible by X, the complex number is itself 

 divisible by 1 — a; and also that if the product of two complex numbers be 

 divisible by 1 —a, one or other of the factors separately must be divisible by 

 1— a. Hence 1— a is a true complex prime, and is the only prime factor of 

 X; infact,X=(l-a)(l-a 2 )...(l-aX-i)= e („)(i-n)^- 1 , if e(a) denote 

 the complex unit 



1-a 2 1-a 3 l-a*-» 



1 — a 1 — a 1 — a 



The theorems which have preceded enable us to give a definition of the 

 norm of an ideal complex number. If the ideal number contain the factor 

 1 — a m times, and if it besides contain k,k',k", .... prime factors of the 

 primes q, q', q", .... appertaining to the exponents/,/',/", .... respectively, 

 we are to understand by its norm, the positive integral number 



\m 9 Vq<k<f'q''k"f" ; 



a definition which, by virtue of the second proposition of this article, is 

 exact in the case of an actually existing number. 



It will be observed that the number of actual or ideal prime factors (com- 

 pound of Xth roots of unity) into which a given real prime can be decom- 

 posed, depends exclusively on the exponent to which the prime appertains 

 for the modulus X. If the exponent is /, the number of ideal factors is 



~ =e. Thus, if q be a primitive root of X, q continues a prime in the 

 J X-i 



complex theory ; if it be a primitive root of the congruence x 2 =1, mod X, 

 it is oidy resoluble into two conjugate prime factors. This dependence of 

 the number of ideal prime factors of a given prime upon the exponent to 

 which it appertains is a remarkable instance of an intimate and simple con- 

 nexion between two properties of the same prime number, which appear at 

 first sight to have no immediate connexion with one another. 



It may be convenient to remark that the word Ideal is sometimes used so 

 as to include, and sometimes so as to exclude, actually existent complex 

 numbers; but it is not apprehended that any confusion can arise from this 

 ambiguity, which it is not worth while to remove at the expense of intro- 

 ducing a new technical term. 



48. Classification of Ideal Numbers. — An ideal number (using the term 

 in its restricted sense) is incapable of being exhibited in an isolated form 

 as a complex integer; as far as has yet appeared, it has no quantitative 

 existence; and the assertion that a given complex number contains an ideal 

 factor, is only a convenient mode of expressing a certain set of congruential 

 conditions which are satisfied by the coefficients of the complex number. 

 Nevertheless we may, without fear of error, represent ideal numbers by the 

 same symbols, /(a), F(a), ^(a) ... , which we have employed to denote 

 actually existing complex numbers, if we are only careful to remember that 

 these symbols, when the numbers which they represent are ideal, admit of 





