On monogenity of certain number fields defined by \(x^8+ax+b\). (English) Zbl 1513.11177
Extending a serious of results on the monogenity of trinomials and monogenity of number fields generated by a root of a trinomial, the author considers trinomials of degree eight. Various conditions are given for the non-monogenity of trinomials and the corresponding number fields.
Among other the author proves: Let \(F(x)=x^8+p^rax+p^kb\in\mathbb{Z}[x]\) with discriminant \(D\) and let \(r\ge k\ge 3\), \(p\nmid ab\), \(k\) odd. Denote by \(D_p\) the \(p\)-free part of \(D\) and assume that \(D_p\) is square-free modulo \(p\).
Let \(K\) be a number field generated by a root of \(F(x)\). Then \(K\) is monogenic, but a root of \(F(x)\) does not generate a power integral basis in \(K\).
Among other the author proves: Let \(F(x)=x^8+p^rax+p^kb\in\mathbb{Z}[x]\) with discriminant \(D\) and let \(r\ge k\ge 3\), \(p\nmid ab\), \(k\) odd. Denote by \(D_p\) the \(p\)-free part of \(D\) and assume that \(D_p\) is square-free modulo \(p\).
Let \(K\) be a number field generated by a root of \(F(x)\). Then \(K\) is monogenic, but a root of \(F(x)\) does not generate a power integral basis in \(K\).
Reviewer: István Gaál (Debrecen)
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; common index divisor; monogeneityReferences:
| [1] | Ahmad, S.; Nakahara, T.; Husnine, SM, Power integral bases for certain pure sextic fields, Internat, J. Number Theory, 10, 2257-2265 (2014) · Zbl 1316.11094 · doi:10.1142/S1793042114500778 |
| [2] | Ahmad, S.; Nakahara, T.; Hameed, A., On certain pure sextic fields related to a problem of Hasse, Internat. J. Algebra Comput., 26, 577-583 (2016) · Zbl 1404.11124 · doi:10.1142/S0218196716500259 |
| [3] | H. Ben Yakkou and L. El Fadil, On monogenity of certain number fields defined by trinomials, arXiv:2109.08765 (2021) · Zbl 1483.11236 |
| [4] | H. Ben Yakkou and L. El Fadil, On monogenity of certain pure number fields defined by \(x^{p^r}-m\), Internat. J. Number Theory (2021), doi:10.1142/S1793042121500858. · Zbl 1483.11236 |
| [5] | H. Ben Yakkou, A. Chillali and L. El Fadil, On power integral bases for certain pure number fields defined by \(x^{2^r \cdot 5^s}- m \), Comm. Algebra, 49, (2021), 2916-2926 · Zbl 1471.11260 |
| [6] | Bilu, Y.; Gaál, I.; Győry, K., Index form equations in sextic fields: a hard computation, Acta Arith., 115, 85-96 (2004) · Zbl 1064.11084 · doi:10.4064/aa115-1-7 |
| [7] | Davis, CT; Spearman, BK, The index of quartic field defined by a trinomial \(x^4+ax+b\), J. Algebra Appl., 17, 1850197 (2018) · Zbl 1437.11149 · doi:10.1142/S0219498818501979 |
| [8] | Dedekind, R., Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Göttingen Abhandlungen, 23, 1-23 (1878) |
| [9] | L. El Fadil, On power integral bases for certain pure number fields defined by \(x^{24}- m \), Studia Sci. Math. Hungar., 57 (2020), 397-407. · Zbl 1474.11182 |
| [10] | El Fadil, L.; Montes, J.; Nart, E., Newton polygons and \(p\)-integral bases of quartic number fields, J. Algebra Appl., 11, 1250073 (2012) · Zbl 1297.11134 · doi:10.1142/S0219498812500739 |
| [11] | A. J. Engler and A. Prestel, Valued Fields, Springer-Verlag (Berlin, Heidelberg, 2005) · Zbl 1128.12009 |
| [12] | Evertse, JH; Győry, K., Discriminant Equations in Diophantine Number Theory (2017), Press: Cambridge Univ, Press · Zbl 1361.11002 · doi:10.1017/CBO9781316160763 |
| [13] | I. Gaál, Diophantine Equations and Power Integral Bases, Theory and Algorithm, 2nd ed., Birkhäuser (Boston, 2019) · Zbl 1465.11090 |
| [14] | Gaál, I., An experiment on the monogenity of a family of trinomials, JP J. Algebra Number Theory Appl., 51, 97-111 (2021) · Zbl 1499.11318 |
| [15] | Gaál, I.; Győry, K., Index form equations in quintic fields, Acta Arith., 89, 379-396 (1999) · Zbl 0930.11091 · doi:10.4064/aa-89-4-379-396 |
| [16] | Gaál, I.; Pethő, A.; Pohst, M., On the resolution of index form equations in quartic number fields, J. Symbolic Comput., 16, 563-584 (1993) · Zbl 0808.11023 · doi:10.1006/jsco.1993.1064 |
| [17] | Gaál, I.; Remete, L., Power integral bases and monogenity of pure fields, J. Number Theory, 173, 129-146 (2017) · Zbl 1419.11118 · doi:10.1016/j.jnt.2016.09.009 |
| [18] | Gaál, I.; Remete, L., Non-monogenity in a family of octic fields, Rocky Mountain J. Math., 47, 817-824 (2017) · Zbl 1381.11102 · doi:10.1216/RMJ-2017-47-3-817 |
| [19] | Gaál, I.; Schulte, N., Computing all power integral bases of cubic number fields, Math. Comput., 53, 689-696 (1989) · Zbl 0677.10013 · doi:10.1090/S0025-5718-1989-0979943-X |
| [20] | Guàrdia, J.; Montes, J.; Nart, E., Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc., 364, 361-416 (2012) · Zbl 1252.11091 · doi:10.1090/S0002-9947-2011-05442-5 |
| [21] | Győry, K., Sur les polynômes à coefficients entiers et de discriminant donné, Publ. Math. Debrecen, 23, 141-165 (1976) · Zbl 0354.10041 |
| [22] | A. Hameed and T. Nakahara, Integral bases and relative monogenity of pure octic fields, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 58, (2015), 419-433 · Zbl 1363.11094 |
| [23] | H. Hasse, Zahlentheorie, Akademie-Verlag (Berlin, 1963) · Zbl 1038.11500 |
| [24] | A. Jakhar and S. Kumar, On non-monogenic number fields defined by \(x^6+ax+b\), Canad. Math. Bull., doi:10.4153/S0008439521000825. · Zbl 1506.11135 |
| [25] | Jhorar, B.; Khanduja, S., On power basis of a class of algebraic number fields, Internat, J. Number Theory, 12, 2317-2321 (2016) · Zbl 1357.11106 · doi:10.1142/S1793042116501384 |
| [26] | Jones, L.; White, D., Monogenic trinomials with non-squarefree discriminant, Internat. J. Math. (2021) · Zbl 1478.11125 · doi:10.1142/S0129167X21500890 |
| [27] | W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer (2004) · Zbl 1159.11039 |
| [28] | Ore, O., Newtonsche Polygone in der Theorie der algebraischen Körper, Math. Ann., 99, 84-117 (1928) · JFM 54.0191.02 · doi:10.1007/BF01459087 |
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