Inteiros p-ádicos e Extensões Ciclotômicas

Abstract

We’ll begin this work with a preliminary Chapter, where we’ll cover introductory concepts of general topology and Galois theory. In Chapter 2, we are going to study Galois groups in more detail. We’ll introduce the Krull topology, a topology that makes any Galois group a topological group. In Chapter 3, we’ll introduce the concepts of inverse limits and profinite groups and show that every Galois group is an example of a profinite group. In Chapter 4, we are going to construct and study the main properties of the field Qp of p-adic numbers. We introduce other metrics on the set of rational numbers and obtain Qp through completion using Cauchy sequences. Finally, in Chapter 5, we’ll study cyclotomic extensions and show that certain infinite cyclotomic extensions of Q are closely related to the multiplicative group of p-adic units.