456 
PRESIDENTIAL ADDRESS TO THE 
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the theory, there first given, of the composition of such forms. It gives also the 
commencement of a like theory of ternary quadratic forms. It contains also the theory 
already referred to, but which has since influenced in so remarkable a manner the 
whole theory of numbers—the theory of the solution of the binomial equation x n — 1 = 0 : 
it is, in fact, the roots or periods of roots derived from these equations which form 
the incommensurables, or unities, of the complex theories which have been chiefly 
worked at; thus, the i of ordinary analysis presents itself as a root of the equation 
¿c 4 — 1 = 0. It was Gauss himself who, for the development of a real theory—that of 
biquadratic residues—found it necessary to use complex numbers of the before-mentioned 
form, ci + bi (a and b positive or negative real integers, including zero), and the theory 
of these numbers was studied and cultivated by Lejeune-Dirichlet. We have thus a 
new theory of these complex numbers, side by side with the former theory of real 
numbers: everything in the real theory reproducing itself, prime numbers, congruences, 
theories of residues, reciprocity, quadratic forms, &c., but with greater variety and 
complexity, and increased difficulty of demonstration. But instead of the equation 
¿r 4 —1=0, we may take the equation ¿e 3 — 1 = 0: we have here the complex numbers 
a + bp composed with an imaginary cube root of unity, the theory specially considered 
by Eisenstein: again a new theory, corresponding to but different from that of the 
numbers a + bi. The general case of any prime value of the exponent n, and with 
periods of roots, which here present themselves instead of single roots, was first con 
sidered by Kummer: viz. if n — l=ef, and 7) 1} are the e periods, each of them 
a sum of f roots, of the equation x n — 1 = 0, then the complex numbers considered 
are the numbers of the form a 1 r] 1 +a 2 rj 2 +... + a e rj e (c¡q, a 2 , ...,a e positive or negative 
ordinary integers, including zero): f may be =1, and the theory for the periods thus 
includes that for the single roots. 
We have thus a new and very general theory, including within itself that of the 
complex numbers a + bi and a + bp. But a new phenomenon presents itself; for these 
special forms the properties in regard to prime numbers corresponded precisely with 
those for real numbers; a non-prime number was in one way only a product of prime 
factors; the power of a prime number has only factors which are lower powers of the 
same prime number: for instance, if p be a prime number, then, excluding the obvious 
decomposition p. _p 2 , we cannot have p 3 = a product of two factors A, B. In the general 
case this is not so, but the exception first presents itself for the number 23; in the 
theory of the numbers composed with the 23rd roots of unity, we have prime 
numbers p, such that p 3 = AB. To restore the theorem, it is necessary to establish 
the notion of ideal numbers; a prime number p is by definition not the product of 
two actual numbers, but in the example just referred to the number p is the product 
of two ideal numbers having for their cubes the two actual numbers A, B, respectively, 
and we thus have p 3 = AB. It is, I think, in this way that we most easily get some 
notion of the meaning of an ideal number, but the mode of treatment (in Kummer’s 
great memoir, “Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten com- 
plexen Zahlen in ihre Primfactoren,” Grelle, t. xxxv. 1847) is a much more refined 
one; an ideal number, without ever being isolated, is made to manifest itself in the 
properties of the prime number of which it is a factor, and without reference to the