Ley de Reciprocidad Cuadr\'atica y aplicaciones

TL;DR

This survey provides a comprehensive, self-contained proof of the Law of Quadratic Reciprocity, illustrating its applications in number theory, elliptic curves, and prime properties, with examples and computational tools included.

Contribution

It offers a rigorous, accessible presentation of quadratic reciprocity and demonstrates its diverse applications with computational examples using GAP.

Findings

Proof of the Law of Quadratic Reciprocity

Applications to prime number properties and elliptic curves

Educational examples with GAP code

Abstract

The goal of this survey is to introduce all the necessary concepts and theorems to provide a rigorous and self-contained proof of the Law of Quadratic Reciprocity and see how this is a useful tool to obtain results such as the problem of the two squares, the problem of determining when a second grade congruence equation has any solutions, to prove some properties about prime numbers and even to obtain information about the zero set of an elliptic curve. We include examples specifically chosen to improve the understanding of the theorems, those examples were created with the program GAP due to its convenience for this topic of mathematics and we include the codes so the reader can readily test it.