This chapter is devoted to the quadratic reciprocity law, proved by Gauss in 1796, when he was only 19 years old. See [118] for a story of the law. We will give two proofs. The second one will depend on Fourier series, and we are therefore going to start the parallel short course in Fourier analysis we talked about in the Introduction.

We now describe a general result on polynomial congruences.

Let f(x) = anxn + … + a0 be a polynomial with integral coefficients, and let p be a prime number. Observe that the solutions of

f(x) ≡ 0 (mod p) (4.1)

are residue classes (mod p). Lagrange proved the following result in 1770.

Theorem 4.1Let f(x) = anxn + an−1xn−1 + … + a1x + a0 be a polynomial with integral coefficients. Let p ∈ P and assume that p ∤ an. Then the congruence

f(x) ≡ 0 (mod p) (4.2)

has at most n solutions (pairwise non-congruent (mod p)).

Proof The case n = 0 is not interesting and the case n = 1 follows from Theorem 3.5. We work by induction and assume that the result is true for polynomials of degree n − 1. Let f(x) have degree n. If (4.2) has no solutions, then the result is true.