Hilding Verdezoto Barros: Modulär aritmetik och kvadratisk reciprocitet

Abstract: This thesis studies fundamental concepts in modular arithmetic, with a particular focus on quadratic residues modulo a prime. After introducing the necessary definitions, Euler’s criterion and the Legendre symbol are presented as tools for determining whether an integer is a quadratic residue modulo an odd prime. A central result of the thesis is the law of quadratic reciprocity, which is treated in detail and proved using a group-theoretic argument following a proof by Rousseau. The final part addresses the problem of computing square roots modulo a prime. For primes with p ≡ 3 (mod 4), a simple explicit formula is given, whereas the case p ≡ 1 (mod 4) requires more advanced methods based on arithmetic in an extended ring. Several examples are provided to illustrate both the theoretical results and the computational methods.