Discriminants of cyclic cubic orders

Let α   be a cubic algebraic integer. Assume that the cubic number field Q(α)Q(α) is Galois. Let α1α1, α2α2 and α3α3 be the real conjugates of α  . We give an explicit ZZ-basis and the discriminant of the Gal(Q(α)/Q)Gal(Q(α)/Q)-invariant totally real cubic order Z[α1,α2,α3]Z[α1,α2,α3]. This new result is completely different from the one previously obtained in the case that the cubic field Q(α)Q(α) is not Galois.