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On power integral bases for certain pure number fields defined by \(x^{24}-m\). (English) Zbl 1474.11182

A number field \(K\) of degree \(n\) is called monogenic if it admits a power integral basis of type \((1,\beta,\ldots,\beta^{n-1})\). Recently integral bases and monogenity of pure number fields of type \(K=\mathbb{Q}(\alpha)\) is intensively investigated, where \(\alpha\) is a root of a monic irreducible polynomial \(x^n-m\) with a square free integer \(m\ne 1\). Several authors characterized monogenity of lower degree pure fields, up to degree 8.
Recently the author extended these results to degree 12, see L. El Fadil [On Power integral bases for certain pure number fields (preprint)]. In the present paper the degree \(n=24\) case is given. The author proves that if \(m\equiv 2\) or \(3\; (\bmod \; 4)\) and \(m\not\equiv \mp 1 \; (\bmod \; 9)\) then \(K\) is monogenic. If \(m\equiv 1\; (\bmod \; 4)\) or \(m\equiv 1\; (\bmod \; 9)\) then \(K\) is not monogenic. The proof uses Newton polygons and the method of Ö. Ore [Math. Ann. 99, 84–117 (1928; JFM 54.0191.02)].

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R16 Cubic and quartic extensions
11R21 Other number fields

Citations:

JFM 54.0191.02
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