NUMBERS. 
614 
[795 
v 7 1 — Tj Q renders actual one of the two ideal factors of any other decomposable number 
3, 13, &c., 
v / 1 — 277 0 \/l - 7] 0 = 1 + Vo, 31 — 12t7o \/i — 7) 0 — — 5 — y 0 , &c. 
Similarly the conjugate multiplier v 7 1 — rji renders actual the other ideal factor of any 
number 2, 3, 13, &c. We have thus two classes, or, reckoning also actual numbers, 
three classes of prime numbers, viz. (i) ideal primes rendered actual by the multiplier 
— Vo, (b) ideal primes rendered actual by the multiplier \/1 — Vi, (iii) actual primes. 
This is a general property in the several complex theories; there is always a finite 
number of classes of ideal numbers, distinguished according to the multipliers by which 
they are rendered actual; the actual numbers form a “ principal ” class. 
40. General theory of congruences—irreducible functions. In the complex theory 
relating to the roots of the equation v 2 + V + 6 = 0, there has just been occasion to con 
sider the equation (v — 4)(t? + 5) = — 2.13, or say the congruence (-?/—4)(77 + 6) = 0 (mod. 13); 
in this form the relation vr + v + 6 = 0 is presupposed, but if, dropping this equation, 
V be regarded as arbitrary, then there is the congruence v 2 + V + 6 = (t? — 4)(7; + 5)(mod. 13). 
For a different modulus, for instance 11, there is not any such congruence exhibiting 
a decomposition of v 2 + V + 6 into factors. The function v 2 + V + 6 is irreducible, that 
is, it is not a product of factors with integer coefficients; in respect of the modulus 
13 it becomes reducible, that is, it breaks up into factors having integer coefficients, 
while for the modulus 11 it continues irreducible. And there is a like general theory 
in regard to any rational and integral function F(x) with integer coefficients; such 
function, assumed to be irreducible, may for a given prime modulus p continue irre 
ducible, that is, it may not admit of any decomposition into factors with integer coefficients ; 
or it may become reducible, that is, admit of a decomposition F(x)=</>(«)^(x)%(#)...(mod.p). 
And, when this is so, it is thus a product, in one way only, of factors <£ (x), ^(x), %(#),..., 
which are each of them irreducible in regard to the same modulus p ; any such factor 
may be a linear function of x, and as such irreducible; or it may be an irreducible 
function of the second or any higher degree. It is hardly necessary to remark that, 
in this theory, functions which are congruent to the modulus p are regarded as identical, 
and that in the expression of F (x) an irreducible function </> (x) may present itself 
either as a simple factor, or as a multiple factor, with any exponent. The decom 
position is analogous to that of a number into its prime factors; and the whole theory 
of the rational and integral function F(x) in regard to the modulus p is in many 
respects analogous to that of a prime number regarded as a modulus. The theory has 
also been studied where the modulus is a power p v . 
41. The congruence-imaginaries of Galois. If F (x) be an irreducible function to 
a given prime modulus p, this implies that there is no integer value of x satisfying 
the congruence F (x) = 0 (mod. p); we assume such a value and call it i, that is, we 
assume F (i) = 0 (mod. p); the step is exactly analogous to that by which, starting from 
the notion of a real root, we introduce into algebra the ordinary imaginary f = V— 1. 
For instance, x 2 — x + 3 is an irreducible function to the modulus 7 : there is no integer 
solution of the congruence x 2 — x + 3 = 0 (mod. 7). Assuming a solution i such that