The positive discriminant case of Nagell's theorem for certain cubic orders (Q626835)

Let \(a\) and \(b\) be two integers and \(f = X^{3} + aX^{2} + bX + 1\) be a cubic polynomial which is irreducible over the rational numbers. Let \(u\) be a real root of \(f\) and \(R:= Z[u]\) be the cubic order generated by \(u\). An element \(c\) of \(R\) is said to be a fundamental unit of \(R\), provided its absolute value \(|c|\) is a member of some basis of the torsion-free part of the unit group of \(R\). A natural question is whether \(u\) is necessary a fundamental unit of \(R\) in this sense. The case where \(f\) has a negative discriminant (case where \(u\) is the unique real root of \(f\)), was treated by \textit{T. Nagell} in his paper [``Zur Theorie der kubischen Irrationalitäten.'' Acta Math. 55, 33--65 (1930; JFM 56.0168.04)]. In this paper, the authors study the case where \(f\) has a positive discriminant (where the other two conjugates of \(u\) are also real); and they prove that \(u\) is always a fundamental unit of the order \(Z[u]\). The exceptions are shown to consist of a single infinite family together with one sporadic case. Their result is analogue of Nagell's result in the case where \(f\) has a negative discriminant (where \(u\) is the unique real root of \(f\)).