Evaluating Zeta Functions of Abelian Number Fields at Negative Integers

In this thesis we study abelian number fields and in particular their zeta functions at the negative integers. The prototypical examples of abelian number fields are the oft-studied cyclotomic fields, a topic upon which many texts have been almost exclusively dedicated to (see for example [26] or nearly any text on global class field theory).;We begin by building up our understanding of the characters of finite abelian groups and how they are related to Dedekind zeta functions. We then use tools from number theory such as the Kronecker-Weber theorem and Bernoulli numbers to find a simple algorithm for determining the values of these zeta functions at negative integers. We conclude the thesis by comparing the relative complexity of our method to two alternative methods that use completely different theoretical tools to attack the more general problem of non-abelian number fields.