  In the past the first named author has studied to what extent the integral
trace can characterize a number field beyond what the discriminant does. The
cases of cyclic number fields and non-totally real fields are more or less
settled, concluding that for such fields the integral trace does not always
characterize the field. In this paper we show that the integral trace is a
complete invariant for degree $n$, $S_n$ real number number fields that satisfy
certain ramification bound. Among the real $S_n$ fields that our results cover,
there are those of square free different ideal. Moreover, for such fields we
find an explicit description of the isometry group of the integral trace.
