The theory of numbers, by Robert D. Carmichael ...

80

THEORY OF NUMBERS

This equation states the law which connects the quadratic
character of q with respect to p with the quadratic character
of p with respect to q. It is known as the Law of Quadratic
Reciprocity. About fifty proofs of it have been given. Its
history has been a very interesting one; see Bachmann's
Niedere Zahlentheorie, Teil I, pp. I80-318, especially pp.
200-206.
For a further account of this beautiful and interesting
subject we refer the reader to Bachmann, loc. cit., and to the
memoirs to which this author gives reference.
~ 42. GALOIS IMAGINARIES
If one is working in the domain of real numbers the equation
X2- I =O
has no solution; for there is no real number whose square is
-i. If, however, one enlarges the "number system" so as
to include not only all real numbers but all complex numbers
as well, then it is true that every algebraic equation has a root.
It is on account of the existence of this theorem for the enlarged
domain that much of the general theory of algebra takes the
elegant form in which we know it.
The question naturally arises as to whether we can make a
similar extension in the case of congruences. The congruence
x2 3 mod 5
has no solution, if we employ the term solution in the sense in
which we have so far used it. But we may if we choose introduce an imaginary quantity, or mark, j such that
j2_ 3 mod 5,
just as in connection with the equation x2+I=o we would
introduce the symbol i having the property expressed by the
equation
i2 = -I.