View More View Less
  • 1 University of Debrecen, H-4002 Debrecen Pf.400, Hungary
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

Abstract

Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type (nm) is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field (nm), where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of (nm) is repeating periodically in m with period length depending on n.

  • [1]

    Berwick, W. E. H., Integral bases, Cambridge University Press (Cambridge Tracts in Mathematics and Mathematical Physics No. 22) (1927).

    • Search Google Scholar
    • Export Citation
  • [2]

    Dedekind, R., Ueber die Anzahl der Idealklassen in reinen kubischen Zahlkörpern, J. Reine Angew. Math., 121 (1900), 40123.

  • [3]

    Funakura, T., On integral bases of pure quartic fields, Math. J. Okayama Univ., 26 (1984), 2741.

  • [4]

    El Fadil, L., Montes, J. and Nart, E., Newton polygons and p-integral bases of quartic number fields, J. Algebra Appl., 11 (4) (2012), 33 p.

    • Search Google Scholar
    • Export Citation
  • [5]

    Gaal, I. and Remete, L., Integral bases and monogenity of pure fields, Journal of Number Theory, 173 (2017), 129146.

  • [6]

    Gaál, I. and Remete, L., Integral bases and monogenity of the simplest sextic fields, Acta Arithmetica, 182 (2) (2018), 173183.

  • [7]

    Gassert, T. A., A note on the monogenity of power maps, Albanian Journal of Mathematics, 11 (2017), 312.

  • [8]

    Guàrdia, J., Montes, J. and Nart, E., Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc., 364 (1) (2012) 361416.

    • Search Google Scholar
    • Export Citation
  • [9]

    Hameed, A. and Nakahara, T., Integral bases and relative monogenity of pure octic fields, Bull. Math. Soc. Sci. Math. Repub. Soc. Roum., Nouv. Sér., 58 (106), (2015) No. 4, 419433.

    • Search Google Scholar
    • Export Citation
  • [10]

    Hameed, A., Nakahara, T., Husnine, S. M. and Ahmad, S., On existence of canonical number system in certain classes of pure algebraic number fields, J. Prime Research in Mathematics, 7 (2011), 1924.

    • Search Google Scholar
    • Export Citation
  • [11]

    Narkiewicz, W., Elementary and analytic theory of algebraic numbers, 3rd ed., Springer Monogr. Math. (2004).

  • [12]

    Okutsu, K., Integral basis of the field ( n a ), Proc. Japan Acad., Ser. A, 58 (1982), 219222.

  • [13]

    Ore, Ø., Newtonsche Polygone in der Theorie der algebraischen Körper, Math. Ann., 99 (1928) 84117.

  • [14]

    Pohst, M. and Zassenhaus, H., Algorithmic algebraic number theory, Cambridge University Press, (1989).

The author instruction is available in PDF.

Please, download the file from HERE

Manuscript submission: HERE

 

  • Impact Factor (2019): 0.486
  • Scimago Journal Rank (2019): 0.234
  • SJR Hirsch-Index (2019): 23
  • SJR Quartile Score (2019): Q3 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.309
  • Scimago Journal Rank (2018): 0.253
  • SJR Hirsch-Index (2018): 21
  • SJR Quartile Score (2018): Q3 Mathematics (miscellaneous)

Language: English, French, German

Founded in 1966
Publication: One volume of four issues annually
Publication Programme: 2020. Vol. 57.
Indexing and Abstracting Services:

  • CompuMath Citation Index
  • Mathematical Reviews
  • Referativnyi Zhurnal/li>
  • Research Alert
  • Science Citation Index Expanded (SciSearch)/li>
  • SCOPUS
  • The ISI Alerting Services

 

Subscribers can access the electronic version of every printed article.

Senior editors

Editor(s)-in-Chief: Pálfy Péter Pál

Managing Editor(s): Sági, Gábor

Editorial Board

  • Biró, András (Number theory)
  • Csáki, Endre (Probability theory and stochastic processes, Statistics)
  • Domokos, Mátyás (Algebra (Ring theory, Invariant theory))
  • Győri, Ervin (Graph and hypergraph theory, Extremal combinatorics, Designs and configurations)
  • O. H. Katona, Gyula (Combinatorics)
  • Márki, László (Algebra (Semigroup theory, Category theory, Ring theory))
  • Némethi, András (Algebraic geometry, Analytic spaces, Analysis on manifolds)
  • Pach, János (Combinatorics, Discrete and computational geometry)
  • Rásonyi, Miklós (Probability theory and stochastic processes, Financial mathematics)
  • Révész, Szilárd Gy. (Analysis (Approximation theory, Potential theory, Harmonic analysis, Functional analysis))
  • Ruzsa, Imre Z. (Number theory)
  • Soukup, Lajos (General topology, Set theory, Model theory, Algebraic logic, Measure and integration)
  • Stipsicz, András (Low dimensional topology and knot theory, Manifolds and cell complexes, Differential topology)
  • Szász, Domokos (Dynamical systems and ergodic theory, Mechanics of particles and systems)
  • Tóth, Géza (Combinatorial geometry)

STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA
Gábor Sági
Address: P.O. Box 127, H–1364 Budapest, Hungary
Phone: (36 1) 483 8344 ---- Fax: (36 1) 483 8333
E-mail: smh.studia@renyi.mta.hu