El Fadil, Lhoussain On power integral bases for certain pure number fields defined by \(x^{24}-m\). (English) Zbl 1474.11182 Stud. Sci. Math. Hung. 57, No. 3, 397-407 (2020). 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)]. Reviewer: István Gaál (Debrecen) Cited in 13 Documents MSC: 11R04 Algebraic numbers; rings of algebraic integers 11R16 Cubic and quartic extensions 11R21 Other number fields Keywords:power integral basis; theorem of Ore; prime ideal factorization Citations:JFM 54.0191.02 PDFBibTeX XMLCite \textit{L. El Fadil}, Stud. Sci. Math. Hung. 57, No. 3, 397--407 (2020; Zbl 1474.11182) Full Text: DOI