On index divisors and monogenity of certain octic number fields defined by $x^8+ax^3+b$

Omar Kchit

doi:10.3842/umzh.v76i7.7536

Omar Kchit Graduate Normal school of Fez, Sidi Mohamed Ben Abdellah University, Morocco

DOI: https://doi.org/10.3842/umzh.v76i7.7536

Keywords: Theorem of Dedekind, Theorem of Ore, prime ideal factorization, Newton polygon, Index of a number field, Power integral basis, Monogenic

Abstract

UDC 511

For any octic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^8+ax^3+b \in \mathbb{Z}[x]$ and for every rational prime $p,$ we show when $p$ divides the index of $K.$ We also describe the prime power decomposition of the index $i(K).$ In this way, we give a partial answer to { Problem $22$} of Narkiewicz [Elementary and analytic theory of algebraic numbers, Springer-Verlag, Auflage (2004)] for this family of number fields. As an application of our results, we conclude that if $i(K)\neq1,$ then $K$ is not monogenic. We illustrate our results by some computational examples.

References

H. Cohen, A course in computational algebraic number theory, GTM, 138, Springer-Verlag, Berlin, Heidelberg (1993). DOI: https://doi.org/10.1007/978-3-662-02945-9

C. T. Davis, B. K. Spearman, The index of a quartic field defined by a trinomial $x^4+ax+b$, J. Algebra and Appl., 17, № 10, 185–197 (2018). DOI: https://doi.org/10.1142/S0219498818501979

R. Dedekind, Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Göttingen Abhandlungen, 23, 1–23 (1878).

L. El Fadil, On common index divisors and monogenity of certain number fields defined by $x^5+ax^2+b$, Comm. Algebra, 50, № 7, 3102–3112 (2022). DOI: https://doi.org/10.1080/00927872.2022.2025820

L. El Fadil, On the index divisors and monogenity of number fields defined $x^5+ax^3+b$, Quaest., 1–11 (2023); DOI: 10.2989/16073606.2022.2156000.

L. El Fadil, I. Gaál, On non-monogenity of certain number fields defined by trinomials $x^4+ax^2+b$ (2022); arXiv:2204.03226. DOI: https://doi.org/10.1515/ms-2023-0063

L. El Fadil, O. Kchit, On index divisors and monogenity of certain {sextic} number fields defined by $x^6+ax^5+b$, Vietnam J. Math. (2024); https://doi.org/10.1007/s10013-023-00679-3. DOI: https://doi.org/10.1007/s10013-023-00679-3

L. El Fadil, O. Kchit, On index divisors and monogenity of certain septic number fields defined by $x^7+ax^3+b$, Comm. Algebra, 1–15 (2022); DOI: 10.1080/00927872.2022.2159035. DOI: https://doi.org/10.5269/bspm.62352

L. El Fadil, J. Montes, E. Nart, Newton polygons and $p$-integral bases of quartic number fields, J. Algebra and Appl., 11, № 4, Article 1250073 (2012). DOI: https://doi.org/10.1142/S0219498812500739

A. J. Engler, Prestel, Valued fields, Springer-Verlag, Berlin, Heidelberg (2005).

H. T. Engstrom, On the common index divisor of an algebraic number field, Trans. Amer. Math. Soc., 32, 223–237 (1930). DOI: https://doi.org/10.1090/S0002-9947-1930-1501535-0

I. Gaál, A. Pethö, M. Pohst, On the indices of biquadratic number fields having Galois group $V_4$, Arch. Math., 57, 357–361 (1991). DOI: https://doi.org/10.1007/BF01198960

J. Guàrdia, J. Montes, E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc., 364, № 1, 361–416 (2012). DOI: https://doi.org/10.1090/S0002-9947-2011-05442-5

K. Hensel, Arithmetische Untersuchungen über die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung, J. reine und angew. Math., 113, 128–160 (1894); DOI: 10.1515/crll.1894.113.128. DOI: https://doi.org/10.1515/crll.1894.113.128

T. Nakahara, On the indices and integral bases of non-cyclic but Abelian biquadratic fields, Arch. Math., 41, № 6, 504–508 (1983). DOI: https://doi.org/10.1007/BF01198579

W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Springer-Verlag, Auflage (2004). DOI: https://doi.org/10.1007/978-3-662-07001-7

E. Nart, On the index of a number field, Trans. Amer. Math. Soc., 289, 171–183 (1985). DOI: https://doi.org/10.1090/S0002-9947-1985-0779058-2

J. Neukirch, Algebraic number theory, Springer-Verlag, Berlin (1999). DOI: https://doi.org/10.1007/978-3-662-03983-0

J. Śliwa, On the nonessential discriminant divisor of an algebraic number field, Acta Arith., 42, 57–72 (1982). DOI: https://doi.org/10.4064/aa-42-1-57-72