On monogenity of certain pure number fields defined by $x^{2^r\cdot7^s}-m$
DOI:
https://doi.org/10.5269/bspm.62352Abstract
Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{2^r\cdot7^s}-m\in \mathbb{Z}[x]$, where $m\neq \pm 1$ is a square free integer, $r$ and $s$ are two positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md{4}$ and $\overline{m}\not\in\{\pm \overline{1},\pm \overline{18},\pm \overline{19}\} \md{49}$, then $K$ is monogenic. But if $r\geq 2$ and $m\equiv 1\md{16}$ or $s\geq 3$, $\overline{m}\in\{ \overline{1}, \overline{18}, -\overline{19}\} \md{49}$, and $\nu_7(m^6-1)\geq 4$, then $K$ is not monogenic. Some illustrating examples are given at the end of the paper.
References
1. Ahmad, S., Nakahara, T., and Husnine, S. M., Power integral bases for certain pure sextic fields, Int. J. Number Theory 10(8), 2257–2265, (2014).
2. Ahmad, S., Nakahara, T., and Husnine, S. M., On certain pure sextic fields related to a problem of Hasse, Int. J. Algebra Comput. 26(3), 577–583, (2016).
3. Ahmad, S., Nakahara, T., and Husnine, S. M.„ On existence of canonical number system in certain classes of pure algebraic number fields, J. Prime Res. Math. 7, 19–24, (2011).
4. Ben Yakkou, H., Chillali, A., and EL Fadil, L., On power integral bases for certain pure number fields defined by x2r·5s− m, Commun. Algebra 49(7), 2916-2926, (2021).
5. Ben Yakkou, H. and El Fadil, L., On monogenity of certain pure number fields defined by xpr− m, Int. J. Number Theory, 17(10), 2235–2242, (2021).
6. Ben Yakkou, H. and Kchit, O., On power integral bases for certain pure number fields defined by x3r− m, S˜ao Paulo J. Math. Sci. (2021).
7. Choulli, H., El Fadil, L., and Kchit, O., On monogenity of certain number fields defined by x60 − m. Acta Math. Vetnam., 1-11 (2022).
8. Cohen, H., A Course in Computational Algebraic Number Theory, GTM 138, Springer-Verlag Berlin Heidelberg (1993).
9. Dedekind, R., Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der hoheren Kongruenzen, Gottingen Abhandlungen 23, 1–23, (1878).
10. El Fadil, L., Computation of a power integral basis of a pure cubic number field, Int. J. Contemp. Math. Sci. 2(13-16), 601–606, (2007).
11. El Fadil, L., On Newton polygon’s techniques and factorization of polynomial over henselian valued fields, J. Algebra its Appl. 19(10), 2050188, (2020).
12. El Fadil, L., On power integral bases for certain pure number fields defined by x12 − m, Pub. Math. 100(1-2), 219-231, (2022).
13. El Fadil, L., On power integral bases for certain pure number fields defined by x24 − m, Studia Sci. Math. Hungarica, 57(3), 397–407, (2020).
14. El Fadil, L., On power integral bases for certain pure number fields defined by x36 − m, Studia Sci. Math. Hungarica, 58(3), 371–380, (2021).
15. El Fadil, L. and Najim, A., On power integral bases for certain pure number fields defined by x2u·3v− m, (2021). arXiv:2106.01252v2
16. El Fadil, L., On power integral bases for certain pure number fields defined by x3u·7v− m, Colloc. Math., (2021).
17. El Fadil, L., On power integral bases for certain pure sextic fields, Bol. Soc. Paran. Math. 40(3s), 1–7, (2022).
18. El Fadil, L., Montes, J., and Nart, E., Newton polygons and p-integral bases of quartic number fields, J. Algebra its Appl. 11(4), 1–33, (2012).
19. Funakura, T., On integral bases of pure quartic fields, Math. J. Okayama Univ. 26, 27-–41, (1984).
20. Gaal, I., Power integer bases in algebraic number fields, Ann. Univ. Sci. Budapest. Sect. Comp. 18, 61–87, (1999).
21. Gaal, I., Diophantine equations and power integral bases, Theory and algorithm, Second edition, Boston, Birkhauser, (2019).
22. Gaal, I. and Gyory, K., Index form equations in quintic fields, Acta Arith. 89, 379–396, (1999).
23. Gaal, I., Olajos, P., Pohst, M., Power integral bases in order of composite fields, Exp. Math. 11(1), 87–90, (2002).
24. Gaal, I. and Remete, L., Binomial Thue equations and power integral bases in pure quartic fields, JP J. Algebra, Number Theory Appl. 32(1), 49–61, (2014).
25. Gaal, I. and Remete, L., Integral bases and monogeneity of pure fields, J. Number Theory 173(1), 129–146, (2018).
26. Guardia, J., Montes, J., and Nart, E., Newton polygons of higher order in algebraic number theory, J. trans. of ams 364(1), 361–416, (2012).
27. Hameed, A. and Nakahara, T., Integral bases and relative monogeneity of pure octic fields, Bull. Math. Soc. Sci. Math. Repub. Soc. Roum. 58(106) No. 4, 419–433, (2015).
28. Hasse, H., Vorlesungen ¨uber Zahlentheorie, Akademie-Verlag, Berlin, (1963).
29. Hensel, K., Arithmetische Untersuchungen ¨uber die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung, J. Reine Angew. Math., 113:128–160, ISSN 0075-4102, (1894).
30. Hensel, K., Theorie der algebraischen Zahlen, Teubner Verlag, Leipzig, Berlin, (1908).
31. Montes, J. and Nart, E., On a theorem of Ore, J. Algebra 146(2), 318–334, (1992).
32. Motoda, Y., Nakahara, T., and Shah, S. I. A., On a problem of Hasse, J. Number Theory 96, 326–334, (2002).
33. Neukirch, J., Algebraic Number Theory, Springer-Verlag, Berlin, (1999).
34. Ore, O., Newtonsche Polygone in der Theorie der algebraischen Korper, Math. Ann. 99, 84–117,(1928).
35. Petho, A. and Pohst, M., On the indices of multiquadratic number fields, Acta Arith. 153(4), 393–414, (2012).
36. Smith, H., The monogeneity of radical extensions. Acta Arith., 198(3), 313-327, (2021).
2. Ahmad, S., Nakahara, T., and Husnine, S. M., On certain pure sextic fields related to a problem of Hasse, Int. J. Algebra Comput. 26(3), 577–583, (2016).
3. Ahmad, S., Nakahara, T., and Husnine, S. M.„ On existence of canonical number system in certain classes of pure algebraic number fields, J. Prime Res. Math. 7, 19–24, (2011).
4. Ben Yakkou, H., Chillali, A., and EL Fadil, L., On power integral bases for certain pure number fields defined by x2r·5s− m, Commun. Algebra 49(7), 2916-2926, (2021).
5. Ben Yakkou, H. and El Fadil, L., On monogenity of certain pure number fields defined by xpr− m, Int. J. Number Theory, 17(10), 2235–2242, (2021).
6. Ben Yakkou, H. and Kchit, O., On power integral bases for certain pure number fields defined by x3r− m, S˜ao Paulo J. Math. Sci. (2021).
7. Choulli, H., El Fadil, L., and Kchit, O., On monogenity of certain number fields defined by x60 − m. Acta Math. Vetnam., 1-11 (2022).
8. Cohen, H., A Course in Computational Algebraic Number Theory, GTM 138, Springer-Verlag Berlin Heidelberg (1993).
9. Dedekind, R., Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der hoheren Kongruenzen, Gottingen Abhandlungen 23, 1–23, (1878).
10. El Fadil, L., Computation of a power integral basis of a pure cubic number field, Int. J. Contemp. Math. Sci. 2(13-16), 601–606, (2007).
11. El Fadil, L., On Newton polygon’s techniques and factorization of polynomial over henselian valued fields, J. Algebra its Appl. 19(10), 2050188, (2020).
12. El Fadil, L., On power integral bases for certain pure number fields defined by x12 − m, Pub. Math. 100(1-2), 219-231, (2022).
13. El Fadil, L., On power integral bases for certain pure number fields defined by x24 − m, Studia Sci. Math. Hungarica, 57(3), 397–407, (2020).
14. El Fadil, L., On power integral bases for certain pure number fields defined by x36 − m, Studia Sci. Math. Hungarica, 58(3), 371–380, (2021).
15. El Fadil, L. and Najim, A., On power integral bases for certain pure number fields defined by x2u·3v− m, (2021). arXiv:2106.01252v2
16. El Fadil, L., On power integral bases for certain pure number fields defined by x3u·7v− m, Colloc. Math., (2021).
17. El Fadil, L., On power integral bases for certain pure sextic fields, Bol. Soc. Paran. Math. 40(3s), 1–7, (2022).
18. El Fadil, L., Montes, J., and Nart, E., Newton polygons and p-integral bases of quartic number fields, J. Algebra its Appl. 11(4), 1–33, (2012).
19. Funakura, T., On integral bases of pure quartic fields, Math. J. Okayama Univ. 26, 27-–41, (1984).
20. Gaal, I., Power integer bases in algebraic number fields, Ann. Univ. Sci. Budapest. Sect. Comp. 18, 61–87, (1999).
21. Gaal, I., Diophantine equations and power integral bases, Theory and algorithm, Second edition, Boston, Birkhauser, (2019).
22. Gaal, I. and Gyory, K., Index form equations in quintic fields, Acta Arith. 89, 379–396, (1999).
23. Gaal, I., Olajos, P., Pohst, M., Power integral bases in order of composite fields, Exp. Math. 11(1), 87–90, (2002).
24. Gaal, I. and Remete, L., Binomial Thue equations and power integral bases in pure quartic fields, JP J. Algebra, Number Theory Appl. 32(1), 49–61, (2014).
25. Gaal, I. and Remete, L., Integral bases and monogeneity of pure fields, J. Number Theory 173(1), 129–146, (2018).
26. Guardia, J., Montes, J., and Nart, E., Newton polygons of higher order in algebraic number theory, J. trans. of ams 364(1), 361–416, (2012).
27. Hameed, A. and Nakahara, T., Integral bases and relative monogeneity of pure octic fields, Bull. Math. Soc. Sci. Math. Repub. Soc. Roum. 58(106) No. 4, 419–433, (2015).
28. Hasse, H., Vorlesungen ¨uber Zahlentheorie, Akademie-Verlag, Berlin, (1963).
29. Hensel, K., Arithmetische Untersuchungen ¨uber die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung, J. Reine Angew. Math., 113:128–160, ISSN 0075-4102, (1894).
30. Hensel, K., Theorie der algebraischen Zahlen, Teubner Verlag, Leipzig, Berlin, (1908).
31. Montes, J. and Nart, E., On a theorem of Ore, J. Algebra 146(2), 318–334, (1992).
32. Motoda, Y., Nakahara, T., and Shah, S. I. A., On a problem of Hasse, J. Number Theory 96, 326–334, (2002).
33. Neukirch, J., Algebraic Number Theory, Springer-Verlag, Berlin, (1999).
34. Ore, O., Newtonsche Polygone in der Theorie der algebraischen Korper, Math. Ann. 99, 84–117,(1928).
35. Petho, A. and Pohst, M., On the indices of multiquadratic number fields, Acta Arith. 153(4), 393–414, (2012).
36. Smith, H., The monogeneity of radical extensions. Acta Arith., 198(3), 313-327, (2021).
Downloads
Published
2022-12-28
Issue
Section
Research Articles
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).
