Asymptotics of Cubic Number Fields with Bounded Second Successive Minimum of the Trace Form

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.