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On Index and Monogenity of Certain Number Fields Defined by Trinomials

  • Lhoussain El Fadil
From the journal Mathematica Slovaca

ABSTRACT

Let K be a number field generated by a root θ of a monic irreducible trinomial F(x)=xn+axm+b[x] . In this paper, we study the problem of monogenity of K. More precisely, we provide some explicit conditions on a, b, n, and m for which K is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree n = 2 r · 3 k with r and k two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this paper.

2020 Mathematics Subject Classification: Primary 11R04; Secondary 11R21; 11Y40

(Communicated by István Gaál)


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 for Professor István Gaál for his encouragement and advice and for Enric Nart who introduced him to Newton polygon techniques.

REFERENCES

[1] AHMAD, S.—NAKAHARA, T.—HAMEED, A: On certain pure sextic fields related to a problem of Hasse, Int. J. Alg. Comput. 26(3) (2016), 577–583.10.1142/S0218196716500259Search in Google Scholar

[2] BEN YAKKOU, H.—EL FADIL, L.: On monogenity of certain number fields defined by trinomials, Funct. Approx. Comment. Math. 67(2) (2022), 199–221.10.7169/facm/1987Search in Google Scholar

[3] CARLITZ, L.: A note on common index divisors, Proc. Amer. Math. Soc. 3 (1952), 688–692.10.1090/S0002-9939-1952-0050627-4Search in Google Scholar

[4] COHEN, H.: A Course in Computational Algebraic Number Theory. GTM 138, Springer-Verlag Berlin Heidelberg, 1993.10.1007/978-3-662-02945-9Search in Google Scholar

[5] DEDEKIND, R.: Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der hUheren Kongruenzen, Göttingen Abhandlungen 23 (1878), 1–23.Search in Google Scholar

[6] DAVIS, C. T.—SPEARMAN, B. K.: The index of a quartic field defined by a trinomial x 4 + ax + b, J. Algebra Appl. 17(10) (2018), Art. ID 1850197.10.1142/S0219498818501979Search in Google Scholar

[7] EL FADIL, L.: On non monogenity of certain number fields defined by a trinomial x 6 + ax 3 + b, J. Number Theory 239 (2022), 489–500.10.1016/j.jnt.2021.10.017Search in Google Scholar

[8] EL FADIL, L.: On common index divisor and monogenity of certain number fields defined by a trinomial x 5 + ax 2 + b, Commun. Algebra 50(7) (2022), 3102–3112.10.1080/00927872.2022.2025820Search in Google Scholar

[9] EL FADIL, L.: On Newton polygon’s techniques and factorization of polynomial over Henselian valued fields, J. Algebra Appl. 19(10) (2020), Art. ID 2050188.10.1142/S0219498820501881Search in Google Scholar

[10] EL FADIL, L.: On power integral bases of certain pure number fields defined by x 3 7 s , Colloq. Math. 169 (2022), 307–317.10.4064/cm8574-6-2021Search in Google Scholar

[11] 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 125073.10.1142/S0219498812500739Search in Google Scholar

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

[13] GAÁL, I.: An experiment on the monogenity of a family of trinomials, JP J. Algebra Number Theory Appl. 51(1) (2021), 97–111.10.17654/NT051010097Search in Google Scholar

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

[15] GUARDIA, 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

[16] HASSE, K.: Zahlentheorie, Akademie-Verlag, Berlin, 1963.10.1515/9783112478202Search in Google Scholar

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

[18] HASSE, K.: Arithmetische Untersuchungen ber Discriminanten und ihre Ausserwesentlichen Theiler, Dissertation, Univ. Berlin, 1884.Search in Google Scholar

[19] 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

[20] JAKHAR, A.—KUMAR, S.: On non-monogenic number fields defined by x 6 + ax + b, Canad. Math. Bull. 65(3) (2022), 788–794.10.4153/S0008439521000825Search in Google Scholar

[21] JAKHAR, A,—KHANDUJA, S.—SANGWAN, N.: Characterization of primes dividing the index of a trinomial, Int. J. Number Theory 13(10) (2017), 2505–2514.10.1142/S1793042117501391Search in Google Scholar

[22] JHORAR, B.—KHANDUJA, S.: On power basis of a class of algebraic number fields, Int. J. Number Theory 12(8) (2016), 2317–2321.10.1142/S1793042116501384Search in Google Scholar

[23] JONES, L.: Infinite families of non-monogenic trinomials, Acta Sci. Math. 87(1–2) (2021), 95–105.10.14232/actasm-021-463-3Search in Google Scholar

[24] JONES, L.: Some new infinite families of monogenic polynomials with non-squarefree discriminant, Acta Arith. 197(2) (2021), 213–219.10.4064/aa200211-21-7Search in Google Scholar

[25] JONES, L.—TRISTAN, PH.: Infinite families of monogenic trinomials and their Galois groups, Int. J. Math. 29(5) (2018), Art. ID 185039.10.1142/S0129167X18500398Search in Google Scholar

[26] JONES, L.—WHITE, D.: Monogenic trinomials with non-squarefree discriminant, Int. J. Math. 32(13) (2021), Art. ID 215089.10.1142/S0129167X21500890Search in Google Scholar

[27] 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

[28] MOTODA, Y.—NAKAHARA, T.—SHAH, S. I. A.: On a problem of Hasse, J. Number Theory 96 (2002), 326–334.10.1006/jnth.2002.2805Search in Google Scholar

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

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

[31] PETHÖ, A.—POHST, M.: On the indices of multiquadratic number fields, Acta Arith. 153(4) (2012), 393–414.10.4064/aa153-4-4Search in Google Scholar

Received: 2022-05-31
Accepted: 2022-09-23
Published Online: 2023-08-04

© 2023 Mathematical Institute Slovak Academy of Sciences

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