Hostname: page-component-669899f699-tzmfd Total loading time: 0 Render date: 2025-04-24T14:30:06.128Z Has data issue: false hasContentIssue false
  • English
  • Français

NORMAL BASES FOR FUNCTION FIELDS

Part of: Discontinuous groups and automorphic forms Field extensions Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  06 May 2024

YOSHINORI HAMAHATA Open the ORCID record for YOSHINORI HAMAHATA [Opens in a new window]
YOSHINORI HAMAHATA*
Affiliation:
Department of Applied Mathematics Okayama University of Science, Ridai-cho 1-1, Okayama 700-0005, Japan
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In function fields in positive characteristic, we provide a concrete example of completely normal elements for a finite Galois extension. More precisely, for a nonabelian extension, we construct completely normal elements for Drinfeld modular function fields using Siegel functions in function fields. For an abelian extension, we construct completely normal elements for cyclotomic function fields.

Keywords

normal basisDrinfeld modular function fieldSiegel functioncyclotomic function field

MSC classification

Primary: 11F52: Modular forms associated to Drinfel'd modules
Secondary: 11G16: Elliptic and modular units 11R60: Cyclotomic function fields (class groups, Bernoulli objects, etc.) 12F05: Algebraic extensions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 1 , August 2024 , pp. 44 - 55
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work was supported by JSPS KAKENHI Grant Number 21K03192.

References

Artin, E., Galois Theory (Dover, New York, 1997).Google Scholar
Bae, S., ‘The arithmetic of Carlitz polynomials’, J. Korean Math. Soc. 35 (1998), 341–260.Google Scholar
Blessenohl, D. and Johnson, K., ‘Eine Verschärfung des Satzes von der Normalbasis’, J. Algebra 103 (1986), 141–159.Google Scholar
Gekeler, E.-U., ‘Modulare Einheiten für Functionenkörper’, J. reine angew. Math. 348 (1984), 94–115.Google Scholar
Gekeler, E.-U., ‘A product expansion for the discriminant function of Drinfeld modules of rank two’, J. Number Theory 21 (1985), 135–140.CrossRefGoogle Scholar
Gekeler, E.-U., Drinfeld Modular Curves, Lecture Notes in Mathematics, 1231 (Springer, Berlin–Heidelberg, 1986).Google Scholar
Goss, D., Basic Structures of Function Fields (Springer, Berlin–Heidelberg, 1996).Google Scholar
Hachenberger, D., ‘Universal normal bases for the abelian closure of the field of rational numbers’, Acta Arith. 93 (2000), 329–341.Google Scholar
Hamahata, Y., ‘Chowla’s theorem over function fields’, Int. J. Number Theory 14 (2018), 1689–1698.Google Scholar
Koo, J. K. and Shin, D. H., ‘Completely normal elements in some finite abelian extensions’, Cent. Eur. J. Math. 11 (2013), 1725–1731.Google Scholar
Koo, J. K., Shin, D. H. and Yoon, D. S., ‘Normal bases for modular function fields’, Bull. Aust. Math. Soc. 95 (2017), 384–392.Google Scholar
Kubert, D. and Lang, S., Modular Units (Springer, New York, 1981).Google Scholar
Leopoldt, H.-W., ‘Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers’, J. reine angew. Math. 201 (1959), 119–149.Google Scholar
Okada, T., ‘On an extension of a theorem of S. Chowla’, Acta Arith. 38 (1980/81), 341–345.Google Scholar
Rosen, M., Number Theory in Function Fields (Springer, New York, 2002).Google Scholar
Schertz, R., ‘Galoismodulstruktur und elliptische Funktionen’, J. Number Theory 39 (1991), 285–326.CrossRefGoogle Scholar