Power bases for cyclotomic integer rings
Let \(p\) be an odd prime, \(\xi\) a primitive \(p\)-th root of unity and \({\mathcal O}_p\) the ring of integers of the cyclotomic field \(\mathbb{Q}(\xi)\). It is a classical problem to investigate power integral bases in \(\mathbb{Q}(\xi)\), that is to determine all \(\alpha\) such that \({\mathcal O}_p=\mathbb{Z}[\alpha]\) or equivalently \(\{ 1,\alpha,\ldots,\alpha^{p-1}\}\) is an integral basis. \textit{A. Bremner} [J. Number Theory 28, 288-298 (1988; Zbl 0637.12001)] conjectured that, up to translations and conjugation only \(\alpha=\xi,1/(1+\xi)\) generate power integral bases. Bremner proved this conjecture for \(p=7\). The author establishes a criterion for Bremner's conjecture for a given regular prime \(p\). The proof of this criterion is based on a determinant formula for the relative class number of \(\mathbb{Q}(\xi)\). This criterion is then used to verify the conjecture for \(p\leq 23\), \(p\neq 17\).
scientific article; zbMATH DE number 1207955