The On power integral bases of certain pure number fields defined by $x^{120}-m $
Abstract
Let $K$ be a pure number field with $\alpha$ a complex root of a monic irreducible polynomial $F(x) = x^{120}-m \in \mathbb{Z}[x]$ with $ m \neq \pm 1 $. In this paper, we study the monogenity of $K$. More precisely, we prove that if $m$ is square-free, $m \not \equiv 1\md{4}$, $m \not \equiv \pm 1 \md{9} $, and $\ol{m}\not \in \{ \mp 1, 7, 18 \} \md {25}$, then $K$ is monogenic. On the other hand, if $m \equiv 1\md{4}$, $m \equiv 1 \md{9} $, or $m \equiv 1 \md{25}$, then $K$ is not monogenic. Our results are illustrated by some computational examples.
Downloads
References
Ahmed, S., Nakahara, T., and Hameed., On certain pure sextic fields related to a problem of Hasse, Int. J. Alg. and Comput. 26(3), 577-583, (2016).
Ahmed, S., Nakahara, T., and Husnine, S. M., Power integral bases for certain pure sextic fields, Int. J. of Number Theory 10(8), 2257-2265, (2014).
Ben Yakkou, H., Chillali, A., and El Fadil, L., On Power integral bases for certain pure number fields defined by x^2r5s - m, Commun. Algebra 49(7), 2916-2926, (2021).
Ben Yakkou, H. and El Fadil, L., On monogenity of certain pure number fields defined by x^pr - m, I. J. of Number theory 17(10), 2235-2242, (2021).
Ben Yakkou, H. and Kchit, O,. On Power integral bases for certain pure number fields defined by x^3r - m, São Paulo J. Math. Sci. 16, 1072-1079, (2022).
Bilu, Y., Gaal, I., and Gyory, K., Index form equations in sextic fields: a hard computation, Acta Arithmetica 115(1), 85-96, (2004).
Cohen, H., A Course in Computational Algebraic Number Theory, GTM 138, Springer-Verlag Berlin Heidelberg, (1993).
Dedekind, R., Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der hoheren Kongruenzen, Gottingen Abhandlungen 23, 1-23, (1878).
El Fadil, L., On Power integral bases for certain pure number fields defined by x^12 -m, Pub. Math Debrecen 100(1-2), 219-231, (2022).
El Fadil, L., On Power integral bases for certain pure number fields defined by x^24 - m, Stud. Sci. Math. Hung. 57(3), 397-407, (2020).
El Fadil, L., Ben Yakkou, H. and Didi, J., On Power integral bases for certain pure number fields defined by x^42 - m, Bol. Soc. Mat. Mex. 27(81), (2021).
El Fadil, L., Montes, J. and Nart, E., Newton polygons and p-integral bases of quartic number fields, J. Algebra and Appl. 11(4), 1-33, (2012).
Evertse, J. H. and Gyory, K., Discriminant equations in diophantine number theory, Cambridge Univ. Press (2017).
Gaal, I., Diophantine equations and power integral bases, Theory and algorithm, 2nd ed., Birkhauser, (Boston, 2019).
Gaal, I. and Gyory, K., Index form equations in quantic fields, Acta Arithmetica 89(4), 379-396, (1999).
Gaal, I. and Remete, L., Binomial Thue equations and power integral bases in pure quartic fields, JP Journal of Algebra Number Theory Appl. 32(1), 49-61, (2014).
Gaal, I. and Remete, L., Power integral bases and monogenity of pure fields, J. of Number Theory 173, 129-146, (2017).
Gaal, I. and Remete, L., Non-monogenity in a family of octic fields, Rocky Mountain J. Math. 47(3), 817-824, (2017).
Gassert, T, A., A note on the monogenity of power maps, A note on the monogenity of power maps 11(1), 3-12, (2017).
Guardia, J., Montes, J. and Nart, E., Newton polygons of higher order in algebraic number theory, Tran. Math. Soc. American 364(1), 361-416, (2012).
Hameed, A. and Nakahara, T., Integral bases and relative monogenity of pure octic fields, Bull. Math. Soc. Sci. Math. R epub. Soc. Roum. 58(106), no. 4, 419-433, (2015).
Hasse, H., Number Theory (English translation), Springer Verlag, (1980).
Narkiewicz, N., Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, Berlin Heidelberg, (2004).
Ore, O., Newtonsche Polygone in der Theorie der algebraischen Korper, Math. Ann.99, 84-117, (1928).
Petho, A. and Pohst, M., On the indices of multiquadratic number fields, Acta Arithmetica 153(4), 393-414, (2012).
Copyright (c) 2024 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
