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

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79

Hence, it follows from (3) that we can readily determine the
quadratic character of m with respect to the odd prime p, that
is, the value of
P\
provided that we know the value of each of the expressions
( P ) ) (P ) ~ (4)
where q is an odd prime.
The first of these can be evaluated at once by means of
Euler's criterion; for, we have
(f (-( jI2) mod p
and hence
Thus we have the following result: The number - i is a quadratic residue of every prime number of the form 4k+ and
a quadratic non-residue of every prime number of the form
4k +3.
The value of the second symbol in (4) is given by the formula
(- =(-I)8(p2_)
The theorem contained in this equation may be stated in the
following words: The number 2 is a quadratic residue of every
prime number of either of the forms 8k +, 8k + 7; it is a quadratic non-residue of every prime number of either of the forms
8k+3, 8k+5.
The proof of this result is not so immediate as that of the
preceding one. To evaluate the third expression in (4) is still
more difficult. We shall omit the demonstration in both of
these cases. For the latter we have the very elegant relation
(P) (p) (- )i(p-i)(s-1)
\ qlp