Abstract: Given a nonzero integer d, we know, by Hermite's Theorem, that there exist only finitely many cubic number fields of discriminant d. A natural question is, how to refine the discriminant in such way that we can tell, when two of these fields are isomorphic. Here we consider the binary quadratic form q_K: Tr_{K/ mathbb{Q}}(x2)|_{O^0_K}, and we show that if d is a positive fundamental discriminant, then the isomorphism class of q_K, as a quadratic form over Z², gives such a refinement.