论文标题
$ p^{\ text {th}} $ abelian Extensions的$ p^{\ text {th}}的Galois模块结构
Galois module structure of $p^{\text{th}}$ power classes of abelian extensions of local fields
论文作者
论文摘要
在本文中，我们描述了$ j = \ mathbf {k}^{\ times}/\ mathbf {k}^{\ times p} $的GALOIS模块的结构，其中$ \ mathbf {k k} $是本地字段$ \ mathbf {k} $ fimitive primitive $ p $ - for unitive unitive for-unity fune，fore $ \ mathbf {k}/\ mathbf {k} $是$ p $ - 元素的abelian扩展名，我们证明$ j $是恒定的约旦类型的模块，稳定的约旦型$ [1]^2 $，在某种程度上，这是J.Mináč和J. Swallow的结果。另外，我们通过计算一些不变式来从我们的证明中获利，这些不变性先前是A. Adem，W。Gao，D。B. Karageuzian和J.Mináč仅以$ p = 2 $而引入的。
In this paper, we describe the Galois module structure of $J=\mathbf{K}^{\times}/\mathbf{K}^{\times p}$, where $\mathbf{K}$ is an extension of a local field $\mathbf{k}$ containing a primitive $p$-th root of unity: for instance, if $\mathbf{K}/\mathbf{k}$ is a $p$-elementary abelian extension, we prove that $J$ is a module of constant Jordan type, with stable Jordan type $[1]^2$, which, in a way, extends the result of J. Mináč and J. Swallow. Also, we take profit from our proof by computing some invariants, which were previously introduced by A. Adem, W. Gao, D. B. Karageuzian and J. Mináč only for $p=2$.
