Um estudo sobre a densidade dos conjuntos de ideais primos

Abstract

In this dissertation, our focus was to develop important theorems about the density of sets of prime ideals with the purpose of establishing tools that can be useful for the study of the set of the twin primes, which has always been my main focus in mathematics. Although it is a long-term goal, we do not intend to treat in this text the application of these tools in the specific problem, leaving it to a later work. In the first chapter, we have structured some general concepts of Number Theory, Algebra and Analysis that will be used throughout the text. We avoid spending a lot of time on these issues, as some of these concepts may already be known to the reader. In second chapter we worked on the theory of p-adic numbers as well as the idea of completion of fields through valuations. It is suficient for the statements presented here that we directly define the concept of valuations, but the p-adic figures present historical relevance in the development of valuation theory, so that we believe that the beauty of the p-adic numbers is a sufficient justification for its addition to the text. In the third chapter, we start working on our first density theorem. Here we constructed the notion of characters for finite abelian groups, L-series, and finally the Arithmetic Progression Theorem itself. In the fourth chapter, we studied the Dedekind zeta function to later develops the Dirichlet's Density Theorem. In the last chapter, we arrived at the Tchebotarëv's Density Theorem. After its demonstration, we reviewed some important applications of it. Most of these results can be found in [7