An important theorem of C. Hermite asserts that any set of algebraic number fields, whose discriminants are bounded in absolute value, must be finite. Properly formulated, a similar theorem holds true for function fields in one variable over a finite constant field. This paper gives a new proof of this result by using an analogue of the geometry of numbers approach due to H. Minkowski in the number field case.
Un théorème important de C. Hermite énonce que tout ensemble de corps de nombres, dont les discriminants sont bornés en valeur absolue, doit être fini. Correctement formulé, un résultat similaire est valable pour des corps de fonctions d’une variable sur un corps de constantes fini. Cet article donne une nouvelle preuve de ce résultat par l’analogie avec l’approche de la « géométrie des nombres » de H. Minkowski dans le cas des corps de nombres.
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@article{JTNB_2017__29_3_799_0,
author = {Rosen, Michael},
title = {A {Geometric} {Proof} of {Hermite{\textquoteright}s} {Theorem} in {Function} {Fields}},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {799--813},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {29},
number = {3},
year = {2017},
doi = {10.5802/jtnb.1001},
language = {en},
url = {https://www.numdam.org/articles/10.5802/jtnb.1001/}
}
TY - JOUR AU - Rosen, Michael TI - A Geometric Proof of Hermite’s Theorem in Function Fields JO - Journal de théorie des nombres de Bordeaux PY - 2017 SP - 799 EP - 813 VL - 29 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://www.numdam.org/articles/10.5802/jtnb.1001/ DO - 10.5802/jtnb.1001 LA - en ID - JTNB_2017__29_3_799_0 ER -
%0 Journal Article %A Rosen, Michael %T A Geometric Proof of Hermite’s Theorem in Function Fields %J Journal de théorie des nombres de Bordeaux %D 2017 %P 799-813 %V 29 %N 3 %I Société Arithmétique de Bordeaux %U https://www.numdam.org/articles/10.5802/jtnb.1001/ %R 10.5802/jtnb.1001 %G en %F JTNB_2017__29_3_799_0
Rosen, Michael. A Geometric Proof of Hermite’s Theorem in Function Fields. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 799-813. doi : 10.5802/jtnb.1001. https://www.numdam.org/articles/10.5802/jtnb.1001/
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