On totally real cubic fields whose unit groups are of type \(\{\theta + r,\theta + s\}\) (Q1283167)

The author determines all totally real cubic number fields possessing a system of fundamental units of the form \((\theta+r, \theta+s)\) for some \(r,s \in \mathbb Z\). For this, he uses the method of \textit{H. J. Godwin} [Proc. Camb. Philos. Soc. 56, 318-321 (1960; Zbl 0116.02802)] in the refined form given by \textit{H. Brunotte} and the reviewer [J. Number Theory 11, 552-559 (1979; Zbl 0408.12005)]. Using this method, he gives also a fresh proof of a result of H. Stender concerning units in fields generated by a root of \(x(x+s)(x+t)-1\) (Theorem 1). As H. J. Godwin observed, this proof can be simplified considerably attending the approximate size of the roots.