Gauss’ Theorema Aureum: The Law of Quadratic Reciprocity

Abstract

Proposition 1.1 of Chap. 1 shows that the solution of the general second-degree congruence ax ² + bx + c ≡ 0 mod p for an odd prime p can be reduced to the solution of the congruence x ² ≡ b ² − 4ac mod p, and we also saw how the solution of x ² ≡ n mod m for a composite modulus m can be reduced by way of Gauss’ algorithm to the solution of x ² ≡ q mod p for prime numbers p and q. In this chapter, we will discuss a remarkable theorem known as the Law of Quadratic Reciprocity, which provides a very powerful method for determining the solvability of congruences of this last type. The theorem states that if p and q are distinct odd primes then the congruences x ² ≡ q mod p and x ² ≡ p mod q are either both solvable or both not solvable, unless p and q are both congruent to 3 mod 4, in which case one is solvable and the other is not. As a simple but no less striking example of the power of this theorem, suppose one wants to know if x ² ≡ 5 mod 103 has any solutions. Since 5 is not congruent to 3 mod 4, the quadratic reciprocity law asserts that x ² ≡ 5 mod 103 and x ² ≡ 103 mod 5 are both solvable or both not. But solution of the latter congruence reduces to x ² ≡ 3 mod 5, which clearly has no solutions. Hence neither does x ² ≡ 5 mod 103.