  In 2012 the first named author conjectured that totally real quartic fields
of fundamental discriminant are determined by the isometry class of the
integral trace zero form; such conjecture was based on computational evidence
and the analog statement for cubic fields which was proved using Bhargava's
higher composition laws on cubes. Here, using Bhargava's parametrization of
quartic fields we prove the conjecture by generalizing the ideas used in the
cubic case. Since at the moment, for arbitrary degrees, there is nothing like
Bhargava's parametrizations we cannot deal with degrees $n > 5$ in a similar
fashion. Nevertheless, using some of our previous work on trace forms we
generalize this result to higher degrees; we show that if $n \ge 3$ is an
integer such that $(\mathbb{Z}/n\mathbb{Z})^{*}$ is a cyclic group, then the
shape is a complete invariant for totally real degree $n$ number fields with
fundamental discriminant.
