Hilbert-Speiser number fields and Stickelberger ideals
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 589-607

Let p be a prime number. We say that a number field F satisfies the condition (H p n ) when any abelian extension N/F of exponent dividing p n has a normal integral basis with respect to the ring of p-integers. We also say that F satisfies (H p ) when it satisfies (H p n ) for all n1. It is known that the rationals satisfy (H p ) for all prime numbers p. In this paper, we give a simple condition for a number field F to satisfy (H p n ) in terms of the ideal class group of K=F(ζ p n ) and a “Stickelberger ideal” associated to the Galois group Gal (K/F). As an application, we give a candidate of an imaginary quadratic field F which has a possibility of satisfying the very strong condition (H p ) for a small prime number p.

Soit p un nombre premier. On dit qu’un corps de nombres F satisfait la condition (H p n ) si toute extension abélienne N/F d’exposant divisant p n possède une base normale d’entiers sur l’anneau des p-entiers. On dit aussi que F satisfait la condition (H p ) s’il satisfait (H p n ) pour tout n1. Il est bien connu que le corps des rationnels satisfait (H p ) pour les nombres premiers p. Dans cet article, nous donnons une condition simple pour qu’un corps de nombres F satisfasse (H p n ) en termes du groupe des classes d’idéaux de K=F(ζ p n ) et d’un “idéal de Stickelberger” associé au groupe de Galois Gal (K/F). Comme application, nous donnons un corps quadratique imaginaire qui pourait vérifier la condition très forte (H p ) pour un petit nombre premier p.

DOI : https://doi.org/10.5802/jtnb.690
Classification:  11R33,  11R18
@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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     pages = {589-607},
     doi = {10.5802/jtnb.690},
     mrnumber = {2605535},
     zbl = {1205.11119},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/JTNB_2009__21_3_589_0/

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