Relative Galois module structure of integers of abelian fields

Nigel P. Byott; Günter Lettl

doi:10.5802/jtnb.160

Nigel P. Byott; Günter Lettl

Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 1, pp. 125-141.

Abstract
Résumé

Let $L / K$ be an extension of algebraic number fields, where $L$ is abelian over $ℚ$ . In this paper we give an explicit description of the associated order $𝒜_{L / K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_{L}$ , the ring of integers of $L$ , is then isomorphic to $𝒜_{L / K}$ . This generalizes previous results of Leopoldt, Chan & Lim and Bley. Furthermore we show that $𝒜_{L / K}$ is the maximal order if $L / K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to $ℚ^{(m^{'})} / K$ , where $m^{'}$ is the conductor of $K$ .

Soit $L / K$ une extension d'un corps de nombres, où $L$ est abélienne sur $ℚ$ . On établit ici une description explicite de l'ordre associé $𝒜_{L / K}$ de cette extension dans le cas où $K$ est un corps cyclotomique, et on démontre que l'anneau des entiers $o_{L}$ de $L$ est isomorphe à $𝒜_{L / K}$ . Cela généralise des résultats antérieurs de Leopoldt, Chan & Lim et Bley. De plus, on montre que $𝒜_{L / K}$ est l'ordre maximal si $L / K$ est une extension cyclique, totalement et sauvagement ramifiée, linéairement disjointe de $ℚ^{(m^{'})} / K$ , où $m^{'}$ désigne le conducteur de $K$ .

MR: 1399950 | Zbl: 0859.11059

DOI: 10.5802/jtnb.160

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     author = {Nigel P. Byott and G\"unter Lettl},
     title = {Relative {Galois} module structure of integers of abelian fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {125--141},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
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     year = {1996},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.160/}
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Nigel P. Byott; Günter Lettl. Relative Galois module structure of integers of abelian fields. Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 1, pp. 125-141. doi : 10.5802/jtnb.160. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.160/

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