Let be a Galois extension of number fields with Gal and with property that the divisors of are non-ramified in . We denote the ring of integers of by and we study as a -module. In particular we show that the fourth power of the (locally free) class of is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of , together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.
Soit une extension galoisienne de corps de nombres où les diviseurs de sont non-ramifiés dans . On note et l’anneau des entiers de . Nous considérons comme -module et nous démontrons que la quatrième puissance de la classe (localement libre) de est la classe triviale. Afin de démontrer ce résultat, nous utilisons la description de Fröhlich des groupes de classes de modules et son représentant pour la classe des . De plus, nous développons une nouvelle méthode de congruences sur les déterminants pour les algèbres des groupes cycliques et nous démontrons des congruences correspondantes pour les sommes de Gauss.
@article{AIF_1980__30_3_11_0, author = {Taylor, Martin J.}, title = {Galois module structure of rings of integers}, journal = {Annales de l'Institut Fourier}, pages = {11--48}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {30}, number = {3}, year = {1980}, doi = {10.5802/aif.791}, mrnumber = {82e:12008}, zbl = {0416.12004}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.791/} }
TY - JOUR AU - Taylor, Martin J. TI - Galois module structure of rings of integers JO - Annales de l'Institut Fourier PY - 1980 SP - 11 EP - 48 VL - 30 IS - 3 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.791/ DO - 10.5802/aif.791 LA - en ID - AIF_1980__30_3_11_0 ER -
Taylor, Martin J. Galois module structure of rings of integers. Annales de l'Institut Fourier, Volume 30 (1980) no. 3, pp. 11-48. doi : 10.5802/aif.791. http://archive.numdam.org/articles/10.5802/aif.791/
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