795] 
NUMBERS. 
611 
Taking then any number x not divisible by p, the N (p) — 1 residues each 
multiplied by x are, to the modulus p, congruent to the series of residues in a 
different order; and we thus have,—say this is Fermat’s theorem for the complex 
theory—— 1 = 0 (mod. p), with all its consequences, in particular, the theory of 
prime roots. 
In the case of a complex modulus such as 3 + 27, the theory is hardly to be 
distinguished from its analogue in the ordinary theorem; a prime root is = 2, and the 
series of powers is 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, for the modulus 3 + 27 as for 
the modulus 13. But for a real prime such as 3 the prime root is a complex 
number; taking it to be =2+7, we have (2 + if — 1 = 0 (mod. 3), and the series of 
powers in fact is 2 + 7, 7, 2 + 27, 2, 1 + 27, 27, 1 + 7, 1, viz. we thus have the system 
of residues (mod. 3). 
We have in like manner a theory of quadratic residues; a Legendrian symbol 
(which, if p, q, are uneven primes not necessarily primary but subject to the 
condition that their imaginary parts are even, denotes +1 or — 1 according as piWv-u 
is =1 or = — 1 (mod. q), so that ^ = +1 or — 1 according as p is or is not a 
residue of q), a law of reciprocity expressed by the very simple form of equation 
, and generally a system of properties such as that which exists in the 
simplex theory. 
The theory of quadratic forms (a, b, c) has been studied in this complex theory; 
the results correspond to those of the simplex theory. 
y 
Y 
A. 
35. The complex theory with the imaginary cube root of unity has also been 
studied; the imaginary element is here 7, = ■§■ (— 1 + V — 3), a root of the equation 
7 2 + 7 + 1 = 0; the form of the complex number is thus a + 67, where a and b are 
any positive or negative integers, including zero. The conjugate number is a + by 2 , 
= a — b — by, and the product (a + by) (a + by 2 ), =a 2 — ab +b 2 , is the norm of each of the 
factors a+by, a+by 2 . The whole theory corresponds very closely to, but is somewhat 
more simple than, that of the complex numbers a + bi. 
36. The last-mentioned theory is a particular case of the complex theory for the 
imaginary Xth roots of unity, X being an odd prime. Here a is determined by the 
| 
equation ——- = 0, that is, a* -1 + a A ~ 2 + ... + a + 1 = 0, and the form of the complex 
number is /(a), = a + ba + col 2 + ... + koi K ~ 2 , where a, b, c, ..., k, are any positive or negative 
integers, including zero. We have X - 1 conjugate forms, viz. /(a), /(a 2 ),..., /(a* -1 ), and 
the product of these is the norm of each of the factors Nf(a), = Nf(<x 2 ), = ..., = iV/(a A_1 ). 
Taking g any prime root of X, <? A_1 -1=0 (mod. X), the roots a, a 2 , ..., a* -1 , may be 
arranged in the order a, a^, a?' 1 ,..., a gK ~ 2 ; and we have thence a grouping of the roots 
in periods, viz. if X — 1 be in any manner whatever expressed as a product of two 
factors, X — l = ef we may with the X-l roots form e periods g 0 , %,..., v e -i, each of 
77—2