On  integral bases and monogenity of  pure octic number fields  with non-square free parameter

Lhoussain El Fadil; István Gaál

doi:10.5269/bspm.67298

Lhoussain El Fadil Sidi Mohamed Ben Abdellah University https://orcid.org/0000-0003-4175-8064
István Gaál

Abstract

In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $\alpha$ of a monic irreducible polynomial $f(x)=x^8-m\in\Z[x]$, it was assumed that $m\neq \pm 1$ is square free. In this paper, we investigate the monogenity of any pure octic number field, without the condition that $m$ is square free. We start by calculating an integral basis of $\Z_K$, the ring of integers of $K$. In particular, we characterize when $\Z_K=\Z[\alpha]$. We give sufficient conditions on $m$, which guarantee that $K$ is not monogenic. We finish the paper by investigating the case when $m=a^u$, $u\in\{1,3,5,7\}$ and $a\neq \mp 1$ is a square free rational integer.

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References

S. Ahmad, T. Nakahara, and A. Hameed, On certain pure sextic fields related to a problem of Hasse, Int. J. Alg. and Comput. 26(3) (2016), 577–583.

S. Ahmad, T. Nakahara and S. M. Husnine, Non-monogenesis of a family of pure sextic fields, Arch. Sci. (Geneva) 65(7) (2012), 42.

S. Ahmad, T. Nakahara, and S. M. Husnine, Power integral bases for certain pure sextic fields, Int. J. of Number Theory 10(8) (2014), 2257– 2265.

B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, S. M. Watt (eds.), MAPLE, Reference Manual, Watcom Publications, Waterloo, Canada, 1988.

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

R. Dedekind, Uber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der hoheren Kongruenzen, Gottingen Abhandlungen 23 (1878), 1–23.

L. El Fadil, On Newton polygon techniques and factorization of polynomial over Henselian valued fields, J. of Algebra and its Appl. 19(10) (2020), 2050188.

L. El Fadil, On Power integral bases for certain pure sextic fields, Bol. Soc. Parana. Math. (3) 40, paper No 143, 7p. (2022).

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

I.Gaal, Diophantine equations and power integral bases, Theory and algorithms, Second edition, Boston, Birkhauser, 2019.

I. Gaal and L. Remete, Binomial Thue equations and power integral bases in pure quartic fields, JP J. Algebra Number Theory Appl. 32 (2014), No. 1, 49–61.

I. Gaal and L. Remete, Power integral bases and monogenity of pure fields, J. of Number Theory 173 (2017) 129–146.

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

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

A. Hameed and T. Nakahara, Integral bases and relative monogeneity of pure octic fields, Bull. Math. Soc. Sci. Math. Repub. Soc. Roum., 58(106) (2015), No. 4, 419-433.

H. Hasse, Zahlentheorie, Akademie-Verlag, Berlin, 1963.

K. Hensel, Arithemetishe untersuchungen uber die gemeinsamen ausserwesentliche Discriminantentheiler einer Gattung, J. Reine Angew Math, 113 (1894), 128–160.

K. Hensel, Theorie der algebraischen Zahlen, Teubner Verlag, Leipzig, Berlin, 1908.

J. Montes and E. Nart, On a theorem of Ore, J. Algebra 146(2) (1992), 318–334.

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

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

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