On index divisors of certain sextic number fields defined by quadrinomials
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Hamid Ben Yakkou
Abstract
Let K be a number field of degree 6 generated by a root of an irreducible quadrinomial
Acknowledgements
The authors are deeply grateful to the referee, whose valuable comments and suggestions have greatly improved the quality of this paper.
References
[1] S. Ahmad, T. Nakahara and A. Hameed, On certain pure sextic fields related to a problem of Hasse, Internat. J. Algebra Comput. 26 (2016), no. 3, 577–583. 10.1142/S0218196716500259Search in Google Scholar
[2] S. Akhtari, Quartic index form equations and monogenizations of quartic orders, Essent. Number Theory 1 (2022), no. 1, 57–72. 10.2140/ent.2022.1.57Search in Google Scholar
[3] L. Alpöge, M. Bhargava and A. Shnidman, A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so, Math. Ann. 388 (2024), no. 4, 4037–4052. 10.1007/s00208-023-02575-0Search in Google Scholar
[4] T. Arnóczki and G. Nyul, On a conjecture concerning the minimal index of pure quartic fields, Publ. Math. Debrecen 104 (2024), no. 3–4, 471–478. 10.5486/PMD.2024.9770Search in Google Scholar
[5]
H. Ben Yakkou,
On nonmonogenic number fields defined by trinomials of type
[6] H. Ben Yakkou, Non-monogenity of certain quintic number fields defined by trinomials, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N. S.) 70 (2024), no. 1, 71–82. 10.47743/anstim.2024.00005Search in Google Scholar
[7] H. Ben Yakkou, On indices and monogenity of quartic number fields defined by quadrinomials, preprint (2024), https://arxiv.org/abs/2401.12782. Search in Google Scholar
[8]
H. Ben Yakkou and B. Boudine,
On the index of the octic number field defined by
[9] A. Bérczes, J.-H. Evertse and K. Győry, Multiply monogenic orders, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), no. 2, 467–497. 10.2422/2036-2145.201107_005Search in Google Scholar
[10] Y. Bilu, I. Gaál and K. Győry, Index form equations in sextic fields: A hard computation, Acta Arith. 115 (2004), no. 1, 85–96. 10.4064/aa115-1-7Search in Google Scholar
[11]
C. T. Davis and B. K. Spearman,
The index of a quartic field defined by a trinomial
[12] R. Dedekind, Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Göttingen Abh. 23 (1878), 1–23. Search in Google Scholar
[13] H. T. Engstrom, On the common index divisors of an algebraic field, Trans. Amer. Math. Soc. 32 (1930), no. 2, 223–237. 10.1090/S0002-9947-1930-1501535-0Search in Google Scholar
[14] J.-H. Evertse, A survey on monogenic orders, Publ. Math. Debrecen 79 (2011), no. 3–4, 411–422. 10.5486/PMD.2011.5150Search in Google Scholar
[15] J.-H. Evertse and K. Győry, Discriminant Equations in Diophantine Number Theory, New Math. Monogr. 32, Cambridge University, Cambridge, 2017. 10.1017/CBO9781316160763Search in Google Scholar
[16] I. Gaál, Diophantine Equations and Power Integral Bases, Birkhäuser/Springer, Cham, 2019. 10.1007/978-3-030-23865-0Search in Google Scholar
[17] I. Gaál and K. Győry, Index form equations in quintic fields, Acta Arith. 89 (1999), no. 4, 379–396. 10.4064/aa-89-4-379-396Search in Google Scholar
[18] J. Guàrdia, J. Montes and E. Nart, Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields, J. Théor. Nombres Bordeaux 23 (2011), no. 3, 667–696. 10.5802/jtnb.782Search in Google Scholar
[19] J. Guàrdia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364 (2012), no. 1, 361–416. 10.1090/S0002-9947-2011-05442-5Search in Google Scholar
[20] K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné, Acta Arith. 23 (1973), 419–426. 10.4064/aa-23-4-419-426Search in Google Scholar
[21] K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné. III, Publ. Math. Debrecen 23 (1976), no. 1–2, 141–165. 10.5486/PMD.1976.23.1-2.23Search in Google Scholar
[22] K. Győry, Corps de nombres algébriques d’anneau d’entiers monogène, Séminaire Delange–Pisot–Poitou, 20e année: 1978/1979. Théorie des nombres, Fasc. 2, Secrétariat Mathématique, Paris (1980), Exp. No. 26. Search in Google Scholar
[23] K. Győry, On discriminants and indices of integers of an algebraic number field, J. Reine Angew. Math. 324 (1981), 114–126. 10.1515/crll.1981.324.114Search in Google Scholar
[24] K. Hensel, Arithmetische Untersuchungen über Discriminanten und ihre ausserwesentlichen Theiler, Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1884. Search in Google Scholar
[25] R. Ibarra, H. Lembeck, M. Ozaslan, H. Smith and K. E. Stange, Monogenic fields arising from trinomials, Involve 15 (2022), no. 2, 299–317. 10.2140/involve.2022.15.299Search in Google Scholar
[26] L. Jones, Infinite families of non-monogenic trinomials, Acta Sci. Math. (Szeged) 87 (2021), no. 1–2, 95–105. 10.14232/actasm-021-463-3Search in Google Scholar
[27] L. Jones, Sextic reciprocal monogenic dihedral polynomials, Ramanujan J. 56 (2021), no. 3, 1099–1110. 10.1007/s11139-020-00310-wSearch in Google Scholar
[28] L. Jones and D. White, Monogenic trinomials with non-squarefree discriminant, Internat. J. Math. 32 (2021), no. 13, Article ID 2150089. 10.1142/S0129167X21500890Search in Google Scholar
[29] P. Llorente and E. Nart, Effective determination of the decomposition of the rational primes in a cubic field, Proc. Amer. Math. Soc. 87 (1983), no. 4, 579–585. 10.1090/S0002-9939-1983-0687621-6Search in Google Scholar
[30] J. Montes and E. Nart, On a theorem of Ore, J. Algebra 146 (1992), no. 2, 318–334. 10.1016/0021-8693(92)90071-SSearch in Google Scholar
[31] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer Monogr. Math., Springer, Berlin, 2004. 10.1007/978-3-662-07001-7Search in Google Scholar
[32] E. Nart, On the index of a number field, Trans. Amer. Math. Soc. 289 (1985), no. 1, 171–183. 10.1090/S0002-9947-1985-0779058-2Search in Google Scholar
[33] J. Odjoumani, A. Togbé and V. Ziegler, On a family of biquadratic fields that do not admit a unit power integral basis, Publ. Math. Debrecen 94 (2019), no. 1–2, 1–19. 10.5486/PMD.2019.8103Search in Google Scholar
[34] Ö. Ore, Newtonsche Polygone in der Theorie der algebraischen Körper, Math. Ann. 99 (1928), no. 1, 84–117. 10.1007/BF01459087Search in Google Scholar
[35] A. Pethő and M. E. Pohst, On the indices of multiquadratic number fields, Acta Arith. 153 (2012), no. 4, 393–414. 10.4064/aa153-4-4Search in Google Scholar
[36] A. Pethő and V. Ziegler, On biquadratic fields that admit unit power integral basis, Acta Math. Hungar. 133 (2011), no. 3, 221–241. 10.1007/s10474-011-0103-5Search in Google Scholar
[37] L. Remete, Integral bases of pure fields with square-free parameter, Studia Sci. Math. Hungar. 57 (2020), no. 1, 91–115. 10.1556/012.2020.57.1.1450Search in Google Scholar
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