Integral trace forms associated to cubic extensions

Given a nonzero integer d, we know by Hermite's Theorem that there exist only finitely many cubic number fields of discriminant d. However, it can happen that two nonisomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form qK : tr(K/Q) (x(2))vertical bar(0)(OK) as such a refinement. For a cubic field of fundamental discriminant d we show the existence of an element T(K) in Bhargava's class group Cl (Z(2) circle times Z(2) circle times Z(2); -3d) such that q(K) is completely determined by T(K). By using one of Bhargava's composition laws, we show that q(K) is a complete invariant whenever K is totally real and of fundamental discriminant.