Let be a prime number. We say that a number field satisfies the condition when any abelian extension of exponent dividing has a normal integral basis with respect to the ring of -integers. We also say that satisfies when it satisfies for all . It is known that the rationals satisfy for all prime numbers . In this paper, we give a simple condition for a number field to satisfy in terms of the ideal class group of and a “Stickelberger ideal” associated to the Galois group . As an application, we give a candidate of an imaginary quadratic field which has a possibility of satisfying the very strong condition for a small prime number .
Soit un nombre premier. On dit qu’un corps de nombres satisfait la condition si toute extension abélienne d’exposant divisant possède une base normale d’entiers sur l’anneau des -entiers. On dit aussi que satisfait la condition s’il satisfait pour tout . Il est bien connu que le corps des rationnels satisfait pour les nombres premiers . Dans cet article, nous donnons une condition simple pour qu’un corps de nombres satisfasse en termes du groupe des classes d’idéaux de et d’un “idéal de Stickelberger” associé au groupe de Galois . Comme application, nous donnons un corps quadratique imaginaire qui pourait vérifier la condition très forte pour un petit nombre premier .
@article{JTNB_2009__21_3_589_0, author = {Ichimura, Humio}, title = {Hilbert-Speiser number fields and {Stickelberger} ideals}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {589--607}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {3}, year = {2009}, doi = {10.5802/jtnb.690}, zbl = {1205.11119}, mrnumber = {2605535}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.690/} }
TY - JOUR AU - Ichimura, Humio TI - Hilbert-Speiser number fields and Stickelberger ideals JO - Journal de théorie des nombres de Bordeaux PY - 2009 SP - 589 EP - 607 VL - 21 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.690/ DO - 10.5802/jtnb.690 LA - en ID - JTNB_2009__21_3_589_0 ER -
%0 Journal Article %A Ichimura, Humio %T Hilbert-Speiser number fields and Stickelberger ideals %J Journal de théorie des nombres de Bordeaux %D 2009 %P 589-607 %V 21 %N 3 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.690/ %R 10.5802/jtnb.690 %G en %F JTNB_2009__21_3_589_0
Ichimura, Humio. Hilbert-Speiser number fields and Stickelberger ideals. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 589-607. doi : 10.5802/jtnb.690. http://archive.numdam.org/articles/10.5802/jtnb.690/
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