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Definition 6.7.1.
Let n be a positive integer,
and let a be an integer such that
n a.
Then a is called a
quadratic residue modulo n
if the congruence
x^{2} a (mod n)
is solvable, and a
quadratic nonresidue
otherwise.
When n is a prime, we write = 1 if a is a quadratic residue modulo n and = -1 if a is a quadratic nonresidue modulo n. The symbol is called the Legendre symbol.
Proposition 6.7.2. [Euler's Criterion]
If p is an odd prime, and if a is in Z with
p a, then
Theorem 6.7.3. [Quadratic Reciprocity]
Let p, q be distinct odd primes. Then
Theorem 6.7.4.
Let p be an odd prime. Then
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