Galois module structure of rings of integers
Annales de l'Institut Fourier, Volume 30 (1980) no. 3, p. 11-48

Let E/F be a Galois extension of number fields with Γ= Gal(E/F) and with property that the divisors of (E:F) are non-ramified in E/Q. We denote the ring of integers of E by 𝒪 E and we study 𝒪 E as a ZΓ-module. In particular we show that the fourth power of the (locally free) class of 𝒪 E 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 E , together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.

Soit E/F une extension galoisienne de corps de nombres où les diviseurs de (E:F) sont non-ramifiés dans E/Q. On note Γ= Gal (E/F) et 𝒪 E l’anneau des entiers de E. Nous considérons 𝒪 E comme ZΓ-module et nous démontrons que la quatrième puissance de la classe (localement libre) de 𝒪 E 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 𝒪 E . 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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {30},
     number = {3},
     year = {1980},
     pages = {11-48},
     doi = {10.5802/aif.791},
     zbl = {0416.12004},
     mrnumber = {82e:12008},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1980__30_3_11_0}
}
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://www.numdam.org/item/AIF_1980__30_3_11_0/

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