an:05519668
Zbl 1228.11006
Castryck, Wouter
A shortened classical proof of the quadratic reciprocity law
EN
Am. Math. Mon. 115, No. 6, 550-551 (2008).
00220216
2008
j
11A15
quadratic reciprocity law; affine varieties; congruences
The author simplifies V.~A.~Lebesgue's proof of the quadratic reciprocity law \((p/q)(q/p) = (-1)^{(p-1)(q-1)/4}\), which was based on counting the number of solutions \(x_1^2 + x_2^2 + \dots + a_n^2 = 1\) over \(\mathbb F_q\). In this article, the number of solutions of \(x_1^2 - x_2^2 + x_3^2 - \dots + a_n^2 = 1\) for odd integers \(n\) is easily computed by induction as \(N_n = q^{n-1} + q^{(n-1)/2}\). Thus \(N_p \equiv 1 + (q/p) \bmod p\) by Fermat and Euler's criterion, and invoking a simple calculation of a multiple Jacobi sum, the reciprocity law follows.
Franz Lemmermeyer (Jagstzell)