On power bases in cyclotomic number fields (Q1098882)

Let p be an odd prime, and \(\zeta^ p=1\neq \zeta\). The author shows that \({\mathbb{Z}}[\zeta]={\mathbb{Z}}[\alpha]\) for \(\alpha =\zeta +\zeta^ 2+...+\zeta^{(p-1)/2}\), and conjectures that every integral generator of the ring \({\mathbb{Z}}[\zeta]\) is of the form \(m\pm \zeta^{\sigma}\) or \(m\pm \alpha^{\sigma}\), where \(m\in {\mathbb{Z}}\) and \(\sigma\) is an automorphism of \({\mathbb{Q}}(\zeta)\). The conjecture is proved to be true for \(p=7\).