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ON THE MONOGENITY OF TOTALLY COMPLEX PURE SEXTIC FIELDS
Abstract
Let $0\neq m \in \mathbb{Z}$ and $\alpha =\sqrt[6]{m}$. According to the results of El Fadil [3], \alpha generates a power integral basis in $K=\mathbb{Q}(\alpha )$, if and only if $m$ is square-free, $m\not\equiv 1(\mod 4)$ and $m\not\equiv \pm 1(\mod 9)$ .
In this note, we consider the case $m < 0$ and present an efficient method to calculate generators of power integral bases in totally complex pure sextic fields. Using this method, we performed an extensive calculation for this type of fields for $0>m>-5000$. In these 1521 fields, we did not find any other (non-equivalent) generators of power integral bases with coefficients in absolute values $<10^{100}$ in the basis $\{1,\alpha ,\alpha ^2,\alpha ^3, \alpha ^4, \alpha ^5\}$.
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I. Gaál, Calculating generators of power integral bases in pure sextic fields, submitted.
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