ON THE THEORY OP NUMBERS, 133 



combination by multiplication or division, but not by addition or subtraction. 

 Thus/(a)x/,(a),/(a)-f-./i(a), [/(a)]"\ are significant symbols, and their 

 interpretation is contained in what has preceded ; but we have no general 

 interpretation of a combination such as/(a)+/i(a)j or /(a)— /(a)*. This 

 symbolic representation of ideal numbers is very convenient, and tends to 

 abbreviate many demonstrations. 



Every ideal number is a divisor of an actual number, and, indeed, of an 

 infinite number of actual numbers. Also, if the ideal number <p(a) be a 

 divisor of the actual number F(a), the quotient 0j(a)=F(a)-r^(a) is always 

 ideal ; for if ^(a) were an actual number, <p(cc), which is the quotient of 

 F(«) divided by (p^a,), ought also to be an actual number. It appears, 

 therefore, that there exists an infinite number of different ideal multipliers, 

 which all render actual the same ideal number. It has, however, been shown 

 by M. Kummer that a finite number of ideal multipliers are sufficient to 

 render actual all ideal numbers whatever; so that it is possible (and that in 

 an infinite number of different ways) to assign a system of ideal multipliers, 

 such that every ideal number is rendered actual by one of them, and one only. 

 Ideal numbers are thus distributed into a certain finite number of classes, — 

 a class comprehending those numbers which are rendered actual by the same 

 multiplier; and this distribution into classes is independent of the particular 

 system of multipliers by which it is effected, inasmuch as it is found that if 

 two ideal numbers be rendered actual by the same multiplier, every other 

 multiplier which renders one of them actual will also render the other actual. 

 Ideal numbers which belong to the same class are said to be equivalent; so 

 that two ideal numbers, which are each of them equivalent to a third, are 

 equivalent to one another. We may regard actual numbers (which need 

 no ideal multiplier) as forming the first or principal class in the distribution, 

 and, consequently, as all equivalent to one another. If/(a) be equivalent to 

 /^a), and </>(a) to ^(a),/(a)xf(a) is equivalent to/, (a) X ^(a), — a result 

 which is expressed by saying that "equivalent ideal numbers multiplied by 

 equivalent numbers, give equivalent products;" and the class of the product 

 is said to be the class compounded of the classes of the factors. 



49. Representation of Ideal Numbers as the roots of Actual Numbers. — 

 An important conclusion is deducible from the theorem that the number of 

 classes of ideal numbers is finite. Let /(a) be any ideal number; and let us 

 consider the series of ideal numbers /(a), /(a) 2 , /(a) 3 , . .. These numbers 

 cannot all belong to different classes ; we can therefore find two different 

 powers of/(a), for example [/(«)]'" and [/(a)3' n+n , which are equivalent 

 to one another. But the equivalence of these numbers implies that [/(a)] n 

 is equivalent to the actual number + 1 ; i. e. that [/(a)]" is itself an actual 

 number. We may therefore enunciate the theorem, " Every ideal number, 

 raised to a certain power, becomes an actual number." 



The index of this power is the same for all ideal numbers of the same class, 

 but may be different for different classes. By reasoning precisely similar 

 to that employed by Euler in his 2nd proof of Fermat's Theoremf, it may 

 be proved that the index of the first term in the series f( a), [/(a)] 2 , 

 [/(a)] 3 . . . , which is an actual number, is either equal to the whole number 

 of classes, or to a submultiple of that number. This least index is said to be 

 the exponent to which the class of ideal numbers containing /(a) appertains. 



* These symbols are, however, interpretablc when /(n) ancl/j(a) belong to the same 

 class. Thus, if ip (a.) X/(«) and <p (a) x/, (a) be both actual, /('*) +f r («) is the ideal quo- 

 tient obtained by dividing ip (a) X/(«) +(p («) Xf x («) by tp (a). 



t See art. 10 of this Report, 



