Generators of the ring of integers of a cyclotomic extension (Q927710)

The author considers cyclotomic fields \({\mathbb Q}(\zeta)\) of prime power order \(q=p^m\). She characterizes power integral bases of \({\mathbb Z} [\zeta]\): If an element \(\alpha\) satisfies \({\mathbb Z} [\alpha]={\mathbb Z} [\zeta]\) then either \(\alpha\) is equivalent to \(\zeta\) or the sum of \(\alpha\) and its complex conjugate \(\bar{\alpha}\) is an odd integer. In this context equivalence means that a Galois image of \(\alpha\) differs from \(\pm \zeta\) by an integer. This improves earlier results by \textit{I. Gaál} and \textit{L. Robertson} [J. Number Theory 120, No. 2, 372--384 (2006; Zbl 1193.11100)] inasmuch an assumption on the class number of \({\mathbb Q}(\zeta +\zeta^{-1})\) is superfluous. The new proof is based on the following lemma. Let \(\alpha\) generate a power integral basis of \({\mathbb Z} [\zeta]\). Denote by \(\sigma_j\) the Galois map satisfying \(\sigma_j (\zeta) =\zeta^j\). Then for \(j\) coprime to \(p\) and \( j \not\equiv 1 \bmod q\) there exists an integer \(b(j,\alpha)\) with \[ \frac{\alpha - \sigma_j(\alpha)}{\bar{\alpha}-\overline{\sigma_j(\alpha)}} \, =\, - \zeta^{b(j,\alpha)}. \]