The p-adic numbers and Hensel's lemma

Abstract: The thesis consists of four chapters, each focused on developing the knowledge and drawing new connections with the p-adic numbers and the notions that are well known from the usual analysis on the real topology. In the first chapter, we introduce the basic structure of the $p$-adics, in connection to the conergent series and field axioms. Following that, in chapter two we focus on the defining the distance in the p-adic world, recalling the definition of the absolute value, valuation and exploring the metric space defined by them. After introducing the distance, we will introduce the p-adic topology connected to it, recall some of the notions from the topology and explore more the p-adic field structure. Our considerations will be rewarded by introducing the Hensel's lemma, centered on polynomials and solving congruence equations with their help.