On the fundamental units of a totally real cubic order generated by a unit

International audienceWe give a new and short proof of J. Beers, D. Henshaw, C. McCall, S. Mulay and M. Spindler following a recent result: if is a totally real cubic algebraic unit, then there exists a unit η ∈ Z such that { , η} is a system of fundamental units of the group UE of the units of the cubic order Z[E], except for an infinite family for which E is a square in Z[E] and one sporadic exception. Not only is our proof shorter, but it enables us to prove a new result: if the conjugates E' and E" of E are in Z[E], then the subgroup generated by E and E' is of bounded index in UE, and if E > 1 > |E'| ≥ |E" | > 0 and if E' and E" are of opposite sign, then { E', E" } is a system of fundamental units of UE.2