Let k be a number field, Ok its ring of integers, Cl(k) its classgroup and h the class number of k. Let Γ be a finite group. Let ℳ be a maximal Ok-order in the semi-simple algebra k[Γ] containing Ok[Γ], and Cl(ℳ) its locally free classgroup. Let A = ℤ ∪{1 m,m ∈ ℤ∖{0, 1,−1}} and n ∈ A. We define the set ℛ(𝒟n,ℳ) of Galois module classes realizable by the nth power of the different to be the set of classes c ∈Cl(ℳ) such that there exists a Galois extension N/k with Galois group isomorphic to Γ (Γ-extension), which is tamely ramified, and for which the class of ℳ⊗Ok[Γ]𝒟N/kn is equal to c, where we clarify that if n = 1 m, where m ∈ ℤ∖{0, 1,−1}, 𝒟N/kn is the |m|th root of the inverse different 𝒟N/k−1 (respectively, the different 𝒟N/k) if m < 0 (respectively, m > 0) when it exists. Let l be a prime number and ξ be a primitive lth root of unity. In this article, we suppose that Γ is cyclic of order l and k/ℚ and ℚ(ξ)/ℚ are linearly disjoint. We prove, sometimes under an assumption on h, that ℛ(𝒟n,ℳ) is a subgroup of Cl(ℳ), by an explicit description using a Stickelberger ideal. In addition, for each n ∈ A, we determine the set of the Steinitz classes of 𝒟N/kn, N/k runs through the tame Γ-extensions of k, and prove that it is a subgroup of Cl(k), also sometimes under an hypothesis on h.

中文翻译：

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Sur la structure galoisienne relative de puissances de la différente et idéaux destickelberger

让ķ是一个数字字段，○ķ它的整数环，氯(ķ)它的班级组和H班级人数ķ. 让Γ是一个有限群。让ℳ成为最大的○ķ- 半简单代数中的阶ķ[Γ]包含○ķ[Γ]， 和氯(ℳ)它的本地免费班级组。让一种 = ℤ ∪{1 米,米 ∈ ℤ∖{0, 1,-1}}和n ∈ 一种. 我们定义集合ℛ(𝒟n,ℳ)可实现的 Galois 模块类n不同的幂是类的集合C ∈氯(ℳ)使得存在一个伽罗瓦扩展ñ/ķ与伽罗瓦群同构Γ(Γ-extension)，它是驯服的分支，并且对于它的类ℳ⊗○ķ[Γ]𝒟ñ/ķn等于C, 我们澄清如果n = 1 米， 在哪里米 ∈ ℤ∖{0, 1,-1},𝒟ñ/ķn是个|米|逆不同的根𝒟ñ/ķ-1（分别地，不同的𝒟ñ/ķ） 如果米 < 0（分别，米 > 0) 当它存在时。让l是一个素数并且ξ做个原始人l统一的根源。在本文中，我们假设Γ是有序循环的l和ķ/ℚ和ℚ(ξ)/ℚ是线性不相交的。我们证明，有时在一个假设下H， 那ℛ(𝒟n,ℳ)是一个子群氯(ℳ)，通过使用 Stickelberger 理想的明确描述。此外，对于每个n ∈ 一种，我们确定了 Steinitz 类的集合𝒟ñ/ķn,ñ/ķ贯穿驯服Γ- 扩展ķ，并证明它是氯(ķ), 有时也是在一个假设下H.