Advertisement

Acta Mathematica Vietnamica

, Volume 44, Issue 3, pp 751–779 | Cite as

Absolutely lq-Finite Extensions

  • El Hassane FliouetEmail author
Article

Abstract

We describe the lower quasi-finite extensions \(K/k\) of characteristic \(p>0\), which are defined as follows: for every \(n\in \mathbb N, k^{p^{-n}} \cap K/k \) is finite. We are especially interested in examining the absolute case. In this regard, we give necessary and sufficient condition for an absolutely lq-finite extension to be of finite size. Moreover, we show that any extension that is at the same time modular and lq-finite is of finite size. Furthermore, we construct an example of extension \(K/k\) of infinite size such that for any intermediate field L of \(K/k, L\) is of finite size over k.

Keywords

Purely inseparable Irrationality degree Modular extension q-finite extension lq-finite extension Absolutely lq-finite extension 

Mathematics Subject Classification (2010)

12F15 

Notes

References

  1. 1.
    Bourbaki, N.: Eléments de Mathématique. Algèbre Commutative. Springer, Berlin (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bourbaki, N.: Algèbre, Chapitre 1 à 3. Springer, Berlin (2007)zbMATHGoogle Scholar
  3. 3.
    Chellali, M., Fliouet, E.H.: Sur les extensions purement inséparable. Arch. Math. (Basel) 81(4), 369–382 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chellali, M., Fliouet, E.H.: Extensions purement inséparables d’exposant non borné. Arch. Math. (Brno) 40(2), 129–159 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chellali, M., Fliouet, E.H.: Extensions presque modulaire. Ann. Sci. Math. Québec 28(1–2), 65–75 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chellali, M., Fliouet, E.H.: Sur la tour des clôtures modulaires. An. Stiint. Univ. “Ovidius” Constanta Ser. Mat. 14(1), 45–66 (2006)MathSciNetGoogle Scholar
  7. 7.
    Chellali, M., Fliouet, E.H.: Théorème de la clôture lq-modulaire et applications. Colloq. Math. 122(2), 275–287 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chellali, M., Fliouet, E.H.: Extensions i-modulaires. Int. J. Algebra 6(9-12), 457–492 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Deveney, J.K.: w 0-generated field extensions. Arch. Math. (Basel) 47(5), 410–412 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fliouet, E.H.: Generalization of quasi-modular extensions. In: Homological and Combinatorial Methods in Algebra, vol. 228, pp. 67–82. Springer International Publishing (2018).  https://doi.org/10.1007/978-3-319-74195-6
  11. 11.
    Fried, M.D., Jarden, M.: Field Arithmetic. Springer, Berlin (2008)zbMATHGoogle Scholar
  12. 12.
    Karpilovsky, G.: Topics in Field Theory. North-Holland Publishing Co., Amsterdam (1989)zbMATHGoogle Scholar
  13. 13.
    Kime, L.A.: Purely inseparable modular extensions of unbounded exponent. Trans. Am. Math. Soc. 176, 335–349 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mordeson, J.N., Vinograde, B.: Structure of Arbitrary Purely Inseparable Extension Fields, p 173. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar
  15. 15.
    Pickert, G.: Inseparable Körperweiterungen. Math. Z. 52, 81–136 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sweedler, M.E.: Structure of inseparable extensions. Ann. Math. (2) 87, 401–410 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Waterhouse, W.C.: The structure of inseparable field extensions. Trans. Am. Math. Soc. 211, 39–56 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Regional Center for the Professions of Education and TrainingAgadirMorocco

Personalised recommendations