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

OTHER TOPICS

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In the remaining three sections we shall give a brief introduction to the theory of Diophantine equations, developing
some of the more elementary properties of certain special
cases. We shall carry this far enough to indicate the nature
of the problem connected with the now famous Last Theorem
of Fermat. The earlier sections of this chapter are not required
as a preliminary to reading this latter part.
~ 4I. THEORY OF QUADRATIC RESIDUES
Let a and m be any two relatively prime integers. In ~ 31
we agreed to say that a is a quadratic residue modulo m or a
quadratic non-residue modulo m according as the congruence
x2==a mod m
has or has not a solution. We saw that if m is chosen equal
to an odd prime number p, then a is a quadratic residue modulo
p or a quadratic non-residue modulo p according as
aa(P-)-I or ar(P-l) --- mod p.
This is known as Euler's criterion.
It is convenient to employ the Legendre symbol
(;)a
to denote the quadratic character of a with respect to p. This
symbol is to have the value + or the value -I according
as a is a quadratic residue modulo p or a quadratic non-residue
modulo p. We shall now derive some of the fundamental properties of this symbol, understanding always that the numbers
in the numerator and the denominator are relatively prime.
From the definition of quadratic residues and non-residues
it is obvious that
()=(p) if a- bmodp. (I)