795] 
NUMBERS. 
593 
positive real prime of the form 4n + 3 is prime, but any positive real prime of the 
form 4a +1 is a sum of two squares, and is thus composite). And disregarding unit 
factors we have, as in the ordinary theory, the theorem that every composite number 
is, in a determinate way, a product of prime factors. 
There is, in like manner, a complex theory involving the cube roots of unity—if 
a be an imaginary cube root of unity (a 2 + a + 1 = 0), then the integers of this theory 
are a + ba (a and b real positive or negative integers, including zero); a complex 
theory with the fifth roots of unity—if a be an imaginary fifth root of unity 
(a 4 + a 3 + a 2 + a + 1 = 0), then the integers of the theory are a + ba + ca? + da? (a, b, c, d, 
real positive or negative integers, including zero); and so on for the roots of the 
orders 7, 11, 13, 17, 19. In all these theories, or at any rate for the orders 3, 5, 7 
(see No. 37, post), we have the foregoing theorem: disregarding unit factors, a number 
other than zero is either prime or composite, and every composite number is, in a 
determinate way, a product of prime factors. But coming to the 23rd roots of unity 
the theorem ceases to be true. Observe that it is a particular case of the theorem 
that, if N be a prime number, any integer power of N has for factors only the lower 
powers of N,—for instance, JS Ts = JV. A 2 ; there is no other decomposition N 3 = AB. 
This is obviously true in the ordinary theory, and it is true in the complex theories 
preceding those for the 3rd, 5th, and 7th roots of unity, and probably in those for 
the other roots preceding the 23rd roots; but it is not true in the theory for the 
23rd roots of unity. We have, for instance, 47, a number not decomposable into factors, 
but 47 3 , =AB, is a product of two numbers each of the form a + ba + ... + kx 21 (a a 
23rd root). The theorem recovers its validity by the introduction into the theory of 
Rummer’s notion of an ideal number. 
The complex theories above referred to would be more accurately described as 
theories for the complex numbers involving the periods of the roots of unity: the 
units are the roots either of the equation x'P -1 + x p ~ 2 + ...+« + 1 = 0 (p a prime number) 
p-1 
or of any equation x e +...±1 = 0 belonging to a factor of the function of the 
order p — 1: in particular, this may be the quadric equation for the periods each of 
roots; they are the theories which were first and have been most completely 
considered, and which led to the notion of an ideal number. But a yet higher 
generalization which has been made is to consider the complex theory, the units 
whereof are the roots of any given irreducible equation which has integer numbers 
for its coefficients. 
There is another complex theory the relation of which to the foregoing is not 
very obvious, viz. Galois’s theory of the numbers composed with the imaginary roots 
of an irreducible congruence, F(x) = 0 (modulus a prime number p); the nature of 
this will be indicated in the sequel. 
In any theory, ordinary or complex, we have a first part, which has been termed 
(but the name seems hardly wide enough) the theory of congruences; a second part, 
the theory of homogeneous forms : this includes in particular the theory of the binary 
quadratic forms (a, b, c)(x, y) 2 ; and a third part, comprising those miscellaneous 
investigations which do not come properly under either of the foregoing heads. 
C. XI. 
75