Nombres $p$-àdics i mètodes local-globals

Abstract

Originally, this project was motivated by the topological and analytical idea of completing a metric space through Cauchy sequences. More specifically, the curiosity came from a willing to study the result of (for a given prime number $p$) completing the set of rational numbers through the $p$-adic norm. The object that arises from this completion of the rational number field is again a field, which we will call the field of the $p$-adic numbers, $\mathbb{Q}_p$. Although it can seem that $p$-adic numbers are an unnatural construction, their field has endless applications in number theory; a part from applications in algebra, topology, quantum mechanics, and complex dynamics. These fields could have many applications in other branches of science (not only in mathematics), which we still have to discover. Furthermore, $\mathbb{Q}_p$ is an object with its own mathematics, since it has been developed a theory of $p$-adic analysis, $p$-adic integration and, more recently, a still arising theory of $p$-adic dynamics.