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A shortened classical proof of the quadratic reciprocity law. (English) Zbl 1228.11006
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.

MSC:
11A15 Power residues, reciprocity
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