The class-number one problem for some real cubic number fields with negative discriminants Original Research Article

We prove that there are effectively only finitely many real cubic number fields of a given class number with negative discriminants and ring of algebraic integers generated by an algebraic unit. As an example, we then determine all these cubic number fields of class number one. There are 42 of them. As a byproduct of our approach, we obtain a new proof of Nagellʹs result according to which a real cubic unit epsilon (Porson)>1 of negative discriminant is generally the fundamental unit of the cubic order Z[epsilon (Porson)].