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Elementary Abelianp-extensions of algebraic function fields

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Abstract

LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderp n.

  1. (a)

    E can be generated overF by some elementy whose minimal polynomial has the specific formT pn−T−z.

  2. (b)

    A formula for the genus ofE is given.

  3. (c)

    IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞).

  4. (d)

    We present a new example of a function fieldE/K whose gap numbers are nonclassical.

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Authors and Affiliations

  1. Pura e Aplicada, Instituto de Matematica, Estrada Dona Castorina 110, 22.460, Rio de Janeiro, Brazil

    Arnaldo Garcia

  2. Fachbereich 6-Mathematik, Universität GHS Essen, D-4300, Essen 1, Fed. Rep. Germany

    Henning Stichtenoth

Authors
  1. Arnaldo Garcia
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  2. Henning Stichtenoth
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Additional information

This paper was written while the first author was visiting Essen within the exchange program GMD-CNPq. The visit was partially supported by the Alexander von Humboldt-Stiftung.

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Garcia, A., Stichtenoth, H. Elementary Abelianp-extensions of algebraic function fields. Manuscripta Math 72, 67–79 (1991). https://doi.org/10.1007/BF02568266

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  • DOI: https://doi.org/10.1007/BF02568266

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