Quadratic Reciprocity and Sign of Gauss Sum via the Finite Weil Representation

Quadratic Reciprocity and the Sign of the Gauss Sum via the Finite Weil Representation

We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p, q and pq. 0. Introduction Two basic results due to Gauss are the quadratic reciprocity law and the sign of th...

Quadratic Reciprocity via Linear Algebra

We adapt a method of Schur to determine the sign in the quadratic Gauss sum and derive from this, the law of quadratic reciprocity.

Gauss Optics and Gauss Sum on an Optical Phenomena

In the previous article (Found Phys. Lett. 16 325-341), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.

Quadratic Reciprocity , after Weil

The character associated to a quadratic extension field K of Q, χ : Z −→ C, χ(n) = (disc(K)/n) (Jacobi symbol), is in fact a Dirichlet character; specifically its conductor is |disc(K)|. This fact encodes basic quadratic reciprocity from elementary number theory, phrasing it in terms that presage class field theory. This writeup discusses Hilbert quadratic reciprocity in the same spirit. Let k ...

A. Quadratic Reciprocity Via Gauss Sums