O anel dos inteiros algébricos de um corpo de números é um domínio de Dedekind.

Abstract

From the point of view of Algebraic Number Theory, a historically important problem was the one of understanding in detail the properties of the ring of algebraic integers of a number field. In this sense, in this work we show that, if A is a Dedekind domain with field of fractions K, if L is a finite separable extension of K and B is the algebraic closure of A in L, then B is also a Dedekind domain. As a consequence of this fact, we conclude that the ring of algebraic integers of a number field is a Dedekind domain, which, in turn, exposes the ubiquity of Dedekind domains. We close the text by characterizing the ring of algebraic integers of the n-th cyclotomic field as the Dedekind domain given by the ring adjunction of the n-th complex roots of unity to the ring Z of integers.