On index divisors and monogenity of certain number fields defined by $$x^{12}+ax^m+b$$

Lhoussain El Fadil1, Omar Kchit1
1Faculty of Sciences Dhar El Mahraz, Sidi Mohamed ben Abdellah University, Atlas-Fes, Morocco

Tóm tắt

Từ khóa


Tài liệu tham khảo

Cohen, H.: A Course in Computational Algebraic Number Theory, GTM 138. Springer-Verlag, Berlin Heidelberg (1993)

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), 185–197 (2018)

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

El Fadil, L.: On common index divisors and monogenity of certain number fields defined by $$x^5+ax^2+b$$. Commun. Algebra (2022). https://doi.org/10.1080/00927872.2022.2025820

El Fadil, L., Gaál, I.: On non-monogenity of certain number fields defined by trinomials $$x^4+ax^2+b$$. http://arxiv.org/abs/2204.03226 (2022)

El Fadil, L., Kchit, O.: On index divisors and monogenity of certain sextic number fields defined by $$x^6+ax^5+b$$. http://arxiv.org/abs/2206.05529 (2022)

El Fadil, L., Kchit, O.: On index divisors and monogenity of certain septic number fields defined by $$x^7+ax^3+b$$. Commun. Algebra 51(6), 2349–2363 (2023)

El Fadil, L., Montes, J., Nart, E.: Newton polygons and $$p$$-integral bases of quartic number fields. J. Algebra Appl. 11(4), 1250073 (2012)

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

Engstrom, H.T.: On the common index divisor of an algebraic number field. Trans. Am. Math. Soc. 32, 223–237 (1930)

Gaál, I.: Diophantine Equations and Power Integral Bases. Theory and Algorithm, 2nd edn. Birkhäuser, Boston (2019)

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

Guardia, J., Montes, J., Nart, E.: Newton polygons of higher order in algebraic number theory. Trans. Am. Math. Soc. 364(1), 361–416 (2012)

Guardia, J., Nart, E.: Genetics of polynomials over local fields. Contemp. Math. 637, 207–241 (2015)

Maclane, S.: A construction for absolute values in polynomial rings. Trans. Am. Math. Soc. 40, 363–395 (1936)

Nakahara, T.: On the indices and integral bases of non-cyclic but abelian biquadratic fields. Arch. Math. 41(6), 504–508 (1983)

Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, 3rd edn. Springer Verlag, Auflag (2004)

Nart, E.: On the index of a number field. Trans. Am. Math. Soc. 289, 171–183 (1985)

Neukirch, J.: Algebraic Number Theory. Springer-Verlag, Berlin (1999)

Ore, O.: Newtonsche Polygone in der Theorie der algebraischen Korper. Math. Ann. 99, 84–117 (1928)

Śliwa, J.: On the nonessential discriminant divisor of an algebraic number field. Acta Arith. 42, 57–72 (1982)