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§ 6.7 Quadratic reciprocity

 
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

(i) , and

(ii) , where k = (p² - 1)/8.

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