The Fundamental System of Units for Cubic Number Fields

Abstract

Let $K$ be a number field of degree $n$. An element $\alpha \in K$ is called integral, if the minimal polynomial of $\alpha$ has integer coefficients. The set of all integral elements of $K$ is denoted by $\mathcal{O}_K$. We will prove several properties of this set, e.g. that $\mathcal{O}_K$ is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group $\mathcal{O}_K^\times$, defined as the set of all invertible elements of $\mathcal{O}_K$. We will prove Dirichlet's Unit Theorem and look at unit groups for the special case of cubic number fields of type $(1,1)$. The structure of the unit group allows us to define a fundamental unit for this type of field. We will study the relation between the discriminant of the number field and this fundamental unit.