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

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 𝒪E l’anneau des entiers de E. Nous considérons 𝒪E comme ZΓ-module et nous démontrons que la quatrième puissance de la classe (localement libre) de 𝒪E 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 𝒪E. 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.

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 𝒪E and we study 𝒪E as a ZΓ-module. In particular we show that the fourth power of the (locally free) class of 𝒪E 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 ℰE, together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.

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     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},
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     url = {https://www.numdam.org/articles/10.5802/aif.791/}
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Taylor, Martin J. Galois module structure of rings of integers. Annales de l'Institut Fourier, Tome 30 (1980) no. 3, pp. 11-48. doi : 10.5802/aif.791. https://www.numdam.org/articles/10.5802/aif.791/

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