Gauss’ Theorema Aureum: The Law of Quadratic Reciprocity

  • Steve Wright
Part of the Lecture Notes in Mathematics book series (LNM, volume 2171)


Proposition  1.1 of Chap.  1 shows that the solution of the general second-degree congruence ax2 + bx + c ≡ 0 mod p for an odd prime p can be reduced to the solution of the congruence x2 ≡ b2 − 4ac mod p, and we also saw how the solution of x2 ≡ n mod m for a composite modulus m can be reduced by way of Gauss’ algorithm to the solution of x2 ≡ 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 x2 ≡ q mod p and x2 ≡ 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 x2 ≡ 5 mod 103 has any solutions. Since 5 is not congruent to 3 mod 4, the quadratic reciprocity law asserts that x2 ≡ 5 mod 103 and x2 ≡ 103 mod 5 are both solvable or both not. But solution of the latter congruence reduces to x2 ≡ 3 mod 5, which clearly has no solutions. Hence neither does x2 ≡ 5 mod 103.


Galois Group Algebraic Number Nonzero Ideal Algebraic Integer Quadratic Residue 
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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Steve Wright
    • 1
  1. 1.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

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