On the fundamental units of some cubic orders generated by units
Résumé
Let ∈ be a totally real cubic algebraic unit. Assume that the cubic number field double-struck Q(∈) is Galois. Let ∈, ∈′ and ∈″ be the three real conjugates of ∈. We tackle the problem of whether {∈, ∈′}is a system of fundamental units of the cubic order double-struck Z[∈, ∈′, ∈″]. Given two units of a totally real cubic order, we explain how one can prove that they form a system of fundamental units of this order. Several explicit families of totally real cubic orders defined by parametrized families of cubic polynomials are considered. We also improve upon and correct several previous results in the literature.