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On the order generated by the conjugates of an algebraic integer

Let α be an algebraic integer of degree n ≥ 3. Assume that the extension Q(α)/Q is Galois. Let Z[conj(α)] be the order of Q(α) generated by the n complex conjugates of α. Apart from the case that Gal(Q(α)/Q) is the symmetric group Sn, only for the cyclic cubic case are an explicit Z-basis and the discriminant of Z[conj(α)] known. Here, we prove that there always exists a Z-basis of Z[conj(α)] containing 1 and α. We deduce a new proof of the cyclic cubic case. We hope that this new approach could be helpful to settle the unsolved Galois quartic case. Finally, for α an algebraic integer of any degree n ≥ 2, it is known that the discriminants of the orders Z[α] go to infinity as k goes to infinity (without assuming that Q(α)/Q is Galois). Then, in the Galois cubic and quartic cases, we propose several conjectures related to the apparent behavior of the orders Z[conj(α)] as k goes to infinity. In particular, the orders Z[conj(α)] seem to behave completely differently from the orders Z[α], as k ≥ 1 goes to infinity.