Teorema de Chebotarev

Abstract

[en] The aim of this dissertation is the study of Chebotarev’s Theorem, which gives the density of a set of prime ideals in terms of conjugacy classes of Frobenius elements. In order to achieve our goal, this work has been divided into three parts. The first part is an introduction to the basic properties of number fields. Moreover, two important examples are studied in detail: quadratic and cyclotomic fields. The aim of this part is to define and describe the ring of integrers of a number field. Furthermore, it is important to understand the Frobenius element to reach the goal of this work, therefore its second part is focused on the Frobenius element associated to a prime ideal of a number field. The element of Frobenius, or more precisely its conjugacy class in the Galois group is essential for the understanding of Chebotarev Theorem. Lastly, assuming previous knowledge corresponding to the Dedekind zeta function, the Chebotarev Theorem is stated and proved, and also two of its most known particular cases are given: the Dirichlet’s Theorem on arithmetic progressions and the Frobenius Theorem.