论文标题
较高本地领域的P延伸的GALOIS组
Galois groups of p-extensions of higher local fields
论文作者
论文摘要
假设$ \ Mathcal k $是$ n $ - 二维特征$ p $，$ \ MATHCAL g = \ MATHOP {gal}（\ Mathcal K_ {sep}/\ Mathcal K）$，$ \ Mathcal G _ {<p} $是$ $ $ $ $ $ $ $ $ $ $ $ $ $ p $ $ \ MATHCAL K _ {<P} \ subset \ Mathcal k_ {sep} $是如此，以至于$ \ Mathop {gal}（\ Mathcal K _ {<p}/\ Mathcal K）= \ Mathcal G _ {<P} $。我们使用Nilpotent Artin-Schreier理论来通过Campbell-Hausdorff的组成法律，从$ g（\ Mathcal l）$组中识别$ \ Mathcal g _ {<p} $（<p} $（\ nathcal l）$。 $ \ Mathcal k $上的Canonical $ \ Mathcal P $ -Topology用于定义$ \ Mathcal l $中的密集谎言subalgebra $ \ Mathcal l^{\ Mathcal p} $。 The algebra $\mathcal L^{\mathcal P}$ can be provided with a system of $\mathcal P$-topological generators and its $\mathcal P$-open subalgebras correspond to all $N$-dimensional extensions of $\mathcal K$ in $\mathcal K_{<p}$.这些结果应用于具有原始$ p $ unity的较高特征0的$ k $。 If $Γ=\mathop{Gal}(K_{alg}/K)$ we introduce similarly the quotient $Γ_{<p}=G(L)$, a dense $\mathbb F_p$-Lie algebra $L^{\mathcal P}\subset L$, and describe the structure of $L^{\mathcal P}$ in terms of generators and关系。通过明确表示$γ_ {<p} $ modulo第三交通剂来说明总体结果。
Suppose $\mathcal K$ is $N$-dimensional local field of characteristic $p$, $\mathcal G =\mathop{Gal}(\mathcal K_{sep}/\mathcal K)$, $\mathcal G_{<p}$ is the maximal quotient of $\mathcal G$ of period $p$ and nilpotent class $<p$ and $\mathcal K_{<p}\subset \mathcal K_{sep}$ is such that $\mathop{Gal}(\mathcal K_{<p}/\mathcal K)=\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to identify $\mathcal G_{<p}$ with the group $G(\mathcal L)$ obtained from a profinite Lie $\mathbb F_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. The canonical $\mathcal P$-topology on $\mathcal K$ is used to define a dense Lie subalgebra $\mathcal L^{\mathcal P}$ in $\mathcal L$. The algebra $\mathcal L^{\mathcal P}$ can be provided with a system of $\mathcal P$-topological generators and its $\mathcal P$-open subalgebras correspond to all $N$-dimensional extensions of $\mathcal K$ in $\mathcal K_{<p}$. These results are applied to higher local fields $K$ of characteristic 0 containing primitive $p$-th root of unity. If $Γ=\mathop{Gal}(K_{alg}/K)$ we introduce similarly the quotient $Γ_{<p}=G(L)$, a dense $\mathbb F_p$-Lie algebra $L^{\mathcal P}\subset L$, and describe the structure of $L^{\mathcal P}$ in terms of generators and relations. The general result is illustrated by explicit presentation of $Γ_{<p}$ modulo third commutators.
