Small generators of function fields
Martin Widmer1
1 Institut für Mathematik A Technische Universität Graz Steyrergasse 30/II 8010 Graz, Austria
Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 747-753.

Soit 𝕂/k une extension finie d’un corps global, donc 𝕂 contient un élément primitif α, c’est à dire 𝕂=k(α). Dans cet article, nous démontrons l’existence d’un élément primitif de petite hauteur dans le cas d’un corps de fonctions. Notre résultat est la réponse pour les corps de fonctions à une question de Ruppert sur les petits générateurs des corps de nombres.

Let 𝕂/k be a finite extension of a global field. Such an extension can be generated over k by a single element. The aim of this article is to prove the existence of a ”small” generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.

DOI : 10.5802/jtnb.744

Martin Widmer 1

1 Institut für Mathematik A Technische Universität Graz Steyrergasse 30/II 8010 Graz, Austria 
@article{JTNB_2010__22_3_747_0,
     author = {Martin Widmer},
     title = {Small generators of function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {747--753},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.744},
     mrnumber = {2769343},
     zbl = {1233.11120},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.744/}
}
TY  - JOUR
AU  - Martin Widmer
TI  - Small generators of function fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2010
SP  - 747
EP  - 753
VL  - 22
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.744/
DO  - 10.5802/jtnb.744
LA  - en
ID  - JTNB_2010__22_3_747_0
ER  - 
%0 Journal Article
%A Martin Widmer
%T Small generators of function fields
%J Journal de théorie des nombres de Bordeaux
%D 2010
%P 747-753
%V 22
%N 3
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.744/
%R 10.5802/jtnb.744
%G en
%F JTNB_2010__22_3_747_0
Martin Widmer. Small generators of function fields. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 747-753. doi : 10.5802/jtnb.744. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.744/

[1] E. Artin Algebraic numbers and algebraic functions, Gordon and Breach, New York, 1967 | MR | Zbl

[2] E. Bombieri; W. Gubler Heights in Diophantine Geometry, Cambridge University Press, 2006 | MR | Zbl

[3] W. Duke Hyperbolic distribution problems and half-integral weight Masss forms, Invent. Math., Volume 92 (1988), pp. 73-90 | MR | Zbl

[4] J. Ellenberg; A. Venkatesh Reflection principles and bounds for class group torsion, Int. Math. Res. Not., Volume no.1, Art. ID rnm002 (2007) | MR | Zbl

[5] K. Mahler An inequality for the discriminant of a polynomial, Michigan Math. J., Volume 11 (1964), pp. 257-262 | MR | Zbl

[6] D. Roy; J. L. Thunder A note on Siegel’s lemma over number fields, Monatsh. Math., Volume 120 (1995), pp. 307-318 | MR | Zbl

[7] W. Ruppert Small generators of number fields, Manuscripta math., Volume 96 (1998), pp. 17-22 | MR | Zbl

[8] J. Silverman Lower bounds for height functions, Duke Math. J., Volume 51 (1984), pp. 395-403 | MR | Zbl

[9] H. Stichtenoth Algebraic function fields and codes, Springer, 1993 | MR | Zbl

[10] J. L. Thunder Siegel’s lemma for function fields, Michigan Math. J., Volume 42 (1995), pp. 147-162 | MR | Zbl

[11] J. D. Vaaler; M. Widmer On small generators of number fields, in preparation (2010)

Cité par Sources :