Algebraic number fields particularly of small degree n have been treated in detail in several
publications during the last years. The subject that has been investigated the most is the
computation of lists of number fields K with field discriminant d(K) less than or equal to a
given bound D and the computation of the minimal value of the discriminant for a given degree n
(and often also signature (r1 r2)) of the number fields. The distinct cases of different
degrees as well as the different numbers of real and complex embeddings respectively are
usually treated independently of each other since each case itself offers a broad set of
problems and questions. In some of the cases the applied methods and algorithms have been
notably improved over the years. Each value for the degree n of the investigated fields
represents a huge and interesting set of problems and questions that can be treated on its own.
The case we will concentrate on in this thesis is n = 3. Algebraic number fields of degree 3
are often referred to as cubic fields and in a way their investigation is easier than the
investigation of higher degree fields since the higher the degree of the field the higher the
number of possible signatures (i.e. combinations of real and complex embeddings of the field).
In this thesis we will concentrate only on totally real cubic fields. Totally real fields are
those fields K for which each embedding of K into the complex numbers C has an image that lies
inside the real numbers R. The purpose of this thesis is to show that the number of isomorphism
classes of cubic fields K whose second successive minima M2(K) as introduced by Minkowski are
less than or equal to a given bound X is asymptotically equal (in X) to the number of cubic
polynomials defining these fields modulo a relation P which will be explained in detail.
