We present a new way of investigating totally real algebraic number
fields of degree 3. Instead of making tables of number fields with
restrictions only on the field discriminant and/or the signature as
described by Pohst, Martinet, Diaz y Diaz, Cohen, and other authors, we
bound not only the field discriminant and the signature but also the
second successive minima of the trace form on the ring of integers O(K)
of totally real cubic fields K. With this, we eventually obtain an
asymptotic behaviour of the size of the set of fields which fulfill the
given requirements. This asymptotical behaviour is only subject to the
bound X for the second successive minima, namely the set in question
will turn out to be of the size O(X^(5/2)). We introduce the necessary
notions and definitions from algebraic number theory, more precisely
from the theory of number fields and from class field theory as well as
some analytical concepts such as (Riemann and Dedekind) zeta functions
which play a role in some of the computations. From the boundedness of
the second successive minima of the trace form of fields we derive
bounds for the coefficients of the polynomials which define those
fields, hence obtaining a finite set of such polynomials. We work out an
elaborate method of counting the polynomials in this set and we show
that errors that arise with this procedure are not of important order.
We parametrise the polynomials so that we have the possibility to apply
further concepts, beginning with the notion of minimality of the
parametrization of a polynomial. Considerations about the consequences
of allowing only minimal pairs (B, C) (as parametrization of a
polynomial f(t)=t^3+at^2+bt+c) to be of interest as well as a bound for
the number of Galois fields among all fields in question and their
importance in the procedure of counting minimal pairs, polynomials, and
fields finally lead to the proof that the number of fields K with second
successive minimum M2(K)
