Home Mathematics On indices of quintic number fields defined by x5 + ax + b
Article
Licensed
Unlicensed Requires Authentication

On indices of quintic number fields defined by x5 + ax + b

  • Lhoussain El Fadil EMAIL logo
Published/Copyright: December 6, 2024
Become an author with De Gruyter Brill

Abstract

The goal of this paper is to calculate explicitly the field index of any quintic number field K generated by a complex root α of a monic irreducible trinomial F(x) = x5 + ax + b ∈ ℤ[x]. In such a way we provide a complete answer to the Problem 22 of Narkiewicz [36] for this class of number fields. Namely, for every prime integer p, we evaluate the highest power of p dividing i(K). In particular, we give sufficient conditions on a and b, which guarantee the non-monogenity of K.

Acknowledgement

The author is deeply grateful to the anonymous referees, whose valuable comments and suggestions have tremendously improved the quality of this paper. As well as to Professor Enric Nart who introduced him to Newton polygon techniques. Also the author thanks Professor István Gaál for his advice and encouragement to work in this area.

  1. Communicated by István Gaál

References

[1] Cohen, H.: A Course in Computational Algebraic Number Theory. Grad. Texts in Math., Vol. 138, Springer-Verlag, Berlin, Heidelberg, 1993.10.1007/978-3-662-02945-9Search in Google Scholar

[2] Davis, C. T.—Spearman, B. K.: The index of a quartic field defined by a trinomial x4 + ax + b, J. Algebra Appl. 17(10) (2018), 185–197.10.1142/S0219498818501979Search in Google Scholar

[3] Dedekind, R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der Thöheren Kongruenzen, Abhandlungen der Kniglichen Gesellschaft der Wissenschaften zu Göttingen 23 (1878), 202–232.Search in Google Scholar

[4] Deajim, A.—El Fadil, L.: On the integral closedness of R[α], Math. Rep. (Bucur.) 24(3) (2022), 571–581.Search in Google Scholar

[5] El Fadil, L.: On integral bases and monogeneity of pure sextic number fields with non-squarefree coefficients, J. Number Theory 228 (2021), 375–389.10.1016/j.jnt.2021.03.025Search in Google Scholar

[6] El Fadil, L.: On common index divisor and monogenity of certain number fields defined by a trinomial x5 + ax2 + b, Comm. Algebra 50(7) (2022), 3102–3112.10.1080/00927872.2022.2025820Search in Google Scholar

[7] El Fadil, L.: On common index divisor and monogenity of certain number fields defined by a trinomial x6 + ax + b, Quaest. Math. 46(8) (2023), 1609–1627.10.2989/16073606.2022.2110537Search in Google Scholar

[8] El Fadil, L.: On the index divisors and monogenity of number fields defined by x5 + ax3 + b, Quaest. Math. 46(11) (2023), 2355–2365.10.2989/16073606.2022.2156000Search in Google Scholar

[9] El Fadil, L.: A note on indices of quartic number fields defined by trinomials x4 + ax + b, Comm. Algebra 52(4) (2024), 1349–1359.10.1080/00927872.2023.2262041Search in Google Scholar

[10] El Fadil, L.: On indices of septic number fields defined by trinomials x7 + ax + b, Mathematics 11(21) (2023), Art. No. 4441.10.3390/math11214441Search in Google Scholar

[11] El Fadil, L.: On non monogenity of certain number fields defined by a trinomial x6 + ax3 + b, J. Number Theory 239 (2022), 489–500.10.1016/j.jnt.2021.10.017Search in Google Scholar

[12] El Fadil, L.—Gaál, I.: On integral bases and monogeneity of pure octic number fields with non-square free parameters; to appear in Bol. Soc. Parana. Mat.Search in Google Scholar

[13] El Fadil, L.—Gaál, I.: On non-monogenity of certain number fields defined by trinomials x4 + ax2 + b, (2022); http://arXiv:2204.03226.10.1515/ms-2023-0063Search in Google Scholar

[14] El Fadil, L.—Kchit, O.: On index divisors and monogenity of certain sextic number fields defined by x6 + ax5 + b, Vietnam J. Math. (2024); https://doi.org/10.1007/s10013-023-00679-3.10.1007/s10013-023-00679-3Search in Google Scholar

[15] El Fadil, L.—Kchit, O.: On index divisors and monogenity of certain septic number fields defined by x7 + ax3 + b, Comm. Algebra 51(6) (2023), 2349–2363.10.1080/00927872.2022.2159035Search in Google Scholar

[16] El Fadil, L.—Kchit, O.: On index divisors and monogenity of certain number fields defined by x12 + axm + b, Ramanujan J. 63 (2024), 451–482.10.1007/s11139-023-00768-4Search in Google Scholar

[17] El Fadil, L.—Montes, J.—Nart, E.: Newton polygons and p-integral bases of quartic number fields, J. Algebra Appl. 11(4) (2012), Art. ID 1250073.10.1142/S0219498812500739Search in Google Scholar

[18] Engstrom, H. T.: On the common index divisor of an algebraic number field, Trans. Amer. Math. Soc. 32(2) (1930), 223–237.10.1090/S0002-9947-1930-1501535-0Search in Google Scholar

[19] Gaál, I.: Diophantine Equations and Power Integral Bases. Theory and Algorithm, 2nd ed., Boston, Birkhäuser, 2019.10.1007/978-3-030-23865-0Search in Google Scholar

[20] Gaál, I.—Pethő, A.—Pohst, M.: On the indices of biquadratic number fields having Galois group V4, Arch. Math. 57 (1991), 357–361.10.1007/BF01198960Search in Google Scholar

[21] Gaál, I.—Pethő, A.—Pohst, M.: On the resolution of index form equations in quartic number fields, J. Symbolic Comput. 16(1993), 563–584.10.1006/jsco.1993.1064Search in Google Scholar

[22] Gaál, I.—Remete, L.: Non-monogenity in a family of octic fields, Rocky Mountain J. Math. 47(3) (2017), 817–824.10.1216/RMJ-2017-47-3-817Search in Google Scholar

[23] Gaál, I.—Remete, L.: Power integral bases and monogenity of pure fields, J. Number Theory 173 (2017), 129–146.10.1016/j.jnt.2016.09.009Search in Google Scholar

[24] Guàrdia, J.—Montes J.—Nart, E.: Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364(1) (2012), 361–416.10.1090/S0002-9947-2011-05442-5Search in Google Scholar

[25] Guàrdia, J.—Nart, E.: Genetics of polynomials over local fields, Contemp. Math. 637 (2015), 207–241.10.1090/conm/637/12767Search in Google Scholar

[26] Hensel, K.: Theorie der Algebraischen Zahlen, Teubner Verlag, Leipzig, Berlin, 1908.Search in Google Scholar

[27] Hensel, K.: Arithmetische Untersuchungen über die gemeinsamen ausserwesentliche Discriminantentheiler einer Gattung, J. Reine Angew Math. 113 (1894), 128–160.10.1515/crll.1894.113.128Search in Google Scholar

[28] Hensel, K.: Untersuchung der Fundamentalgleichung einer Gattung für eine reelle Primzahl als Modul und Bestimmung der Theiler ihrer Discriminante, J. Reine Angew Math. 113 (1894), 61–83.10.1515/crll.1894.113.61Search in Google Scholar

[29] Hensel, K.: Arithmetische Untersuchungen ber Discriminanten und ihre ausserwesentlichen Theiler, Dissertation, Univ. Berlin, 1884.Search in Google Scholar

[30] Ibarra, R.—Lembeck, H.—Ozaslan, M.—Smith, H.—Stange, K. E.: Monogenic fields arising from trinomials, Involve 15(2) (2022), 299–317.10.2140/involve.2022.15.299Search in Google Scholar

[31] Llorente, P.—Nart, E.—Vila, N.: Decomposition of primes in number fields defined by trinomials, J. Théor. Nombres Bordeaux 3 (1991), 27–41.10.5802/jtnb.40Search in Google Scholar

[32] MacLane, S.: A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363–395.10.1090/S0002-9947-1936-1501879-8Search in Google Scholar

[33] Montes, J.—Nart, E.: On a theorem of Ore, J. Algebra 146(2) (1992), 318–334.10.1016/0021-8693(92)90071-SSearch in Google Scholar

[34] Nakahara, T.: On the indices and integral bases of non-cyclic but abelian biquadratic fields, Arch. Math. 41(6) (1983), 504–508.10.1007/BF01198579Search in Google Scholar

[35] Nart, E.: On the index of a number field, Trans. Amer. Math. Soc. 289 (1985), 171–183.10.1090/S0002-9947-1985-0779058-2Search in Google Scholar

[36] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, Springer Verlag, 3rd ed., 2004.10.1007/978-3-662-07001-7Search in Google Scholar

[37] Neukirch, J.: Algebraic Number Theory, Springer-Verlag, Berlin, 1999.10.1007/978-3-662-03983-0Search in Google Scholar

[38] Ore, O.: Newtonsche Polygone in der Theorie der algebraischen Korper, Math. Ann. 99 (1928), 84–117.10.1007/BF01459087Search in Google Scholar

Received: 2023-08-16
Accepted: 2024-08-16
Published Online: 2024-12-06
Published in Print: 2024-12-15

© 2024 Mathematical Institute Slovak Academy of Sciences

Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2024-0102/pdf
Scroll to top button