Abstract
Let \(K = \mathbb {Q} (\alpha )\) be a pure number field generated by a complex root \(\alpha\) of a monic irreducible polynomial \(F(x) = x^{20}-m\), with \(m \ne \mp 1\) a square free rational integer. In this paper, we study the monogenity of K. We prove that if \(m\not \equiv 1\ \text{(mod } {4})\) and \(\overline{m}\not \in \{\overline{1}, \overline{7}, \overline{18}, \overline{24}\} \ \text{(mod } {25})\), then K is monogenic. But if \(m\equiv 1\ \text{(mod } {16})\) or \(m\equiv 1\ \text{(mod } {25})\), then K is not monogenic. Some illustrating examples are given too.
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The author is deeply grateful for the anonymous referee for his careful checking. As well as for Professor Enric Nart who introduced him to Newton polygon techniques.
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Communicated by Julio Andrade.
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Fadil, L.E. On monogenity of certain pure number fields defined by \(x^{20}-m\). São Paulo J. Math. Sci. 16, 1063–1071 (2022). https://doi.org/10.1007/s40863-021-00254-z
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DOI: https://doi.org/10.1007/s40863-021-00254-z