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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 x2 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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