When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Let e be an algebraic unit of the degree n ⩾ 3. Assume that the extension ℚ(e)/ℚ is Galois. We would like to determine when the order ℤ[e] of ℚ(e) is Gal(ℚ(e)/ℚ)-invariant, i.e. when the n complex conjugates e1, …, en of e are in ℤ[e], which amounts to asking that ℤ[e1, …, en] = ℤ[e], i.e., that these two orders of ℚ(e) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order ℤ[e1, e2, e3]. However, there is no known similar formula for n > 3. In the present paper, we put forward and motivate three conjectures for the solution to this determination for n = 4 (two possible Galois groups) and n = 5 (one possible Galois group). In particular, we conjecture that there are only finitely many cyclic quartic and quintic Galois-invariant orders generated by an algebraic unit. As a consequence of our work, we found a parametrized family of monic quartic polynomials in ℤ[X] whose roots e generate bicyclic biquadratic extensions ℚ(e)/ℚ for which the order ℤ[e] is Gal(ℚ(e)/ℚ)-invariant and for which a system of fundamental units of ℤ[e] is known. According to the present work it should be difficult to find other similar families than this one and the family of the simplest cubic fields.