Small generators of function fields
Martin Widmer
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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.744 
@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},
     zbl = {1233.11120},
     mrnumber = {2769343},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.744/}
}
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 237460 | Zbl 0194.35301

[2] E. Bombieri; W. Gubler Heights in Diophantine Geometry, Cambridge University Press, 2006 | MR 2216774 | Zbl 1115.11034

[3] W. Duke Hyperbolic distribution problems and half-integral weight Masss forms, Invent. Math., Volume 92 (1988), pp. 73-90 | MR 931205 | Zbl 0628.10029

[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 2331900 | Zbl 1130.11060

[5] K. Mahler An inequality for the discriminant of a polynomial, Michigan Math. J., Volume 11 (1964), pp. 257-262 | MR 166188 | Zbl 0135.01702

[6] D. Roy; J. L. Thunder A note on Siegel’s lemma over number fields, Monatsh. Math., Volume 120 (1995), pp. 307-318 | MR 1363143 | Zbl 0839.11011

[7] W. Ruppert Small generators of number fields, Manuscripta math., Volume 96 (1998), pp. 17-22 | MR 1624340 | Zbl 0899.11063

[8] J. Silverman Lower bounds for height functions, Duke Math. J., Volume 51 (1984), pp. 395-403 | MR 747871 | Zbl 0579.14035

[9] H. Stichtenoth Algebraic function fields and codes, Springer, 1993 | MR 1251961 | Zbl 0816.14011

[10] J. L. Thunder Siegel’s lemma for function fields, Michigan Math. J., Volume 42 (1995), pp. 147-162 | MR 1322196 | Zbl 0830.11024

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