On power integral bases of certain pure number fields defined by \(x^{3^r} - m\)
The authors consider fields of the form \(K=\mathbb Q(\alpha)\), where \(\alpha=\sqrt[3^r] m\) with \(r\ge1\) and square-free \(m\ne 0,\pm1\). Theorem 1 shows that if \(m\not\equiv\pm1\bmod 9\), then \(K\) is monogenic with \(1,\alpha,\dots,\alpha^{3^r-1}\) forming an integral basis. This result is known, being a particular case of the theorem of [\textit{T. Alden Gassert}, Albanian J. Math. 11, 3--12 (2017; Zbl 1392.11082)]. In Theorem 2 a result of [\textit{Ö. Ore}, Math. Ann. 99, 84--117 (1928; JFM 54.0191.02)] is applied to deduce that if \(r\ge3\) and \(m\equiv\pm1\bmod 81\), then the prime \(3\) is a common index divisor in \(K\), hence \(K\) is not monogenic.
scientific article; zbMATH DE number 7626138
scientific article