Lyndon-Demushkin 方法与 $G_2$ 值 Galois 表示的晶体提升

作为绝对伽罗瓦群的 Demushkin 群的自由积

Free product of Demushkin groups as absolute Galois group

本文证明了可数多个秩为 $\aleph_0$ 的 Demushkin 群在特定条件下的自由 profinite 积构成绝对伽罗瓦群的充分必要条件是 $\log_pq(G_i)\to \infty$。

不可数秩的Demushkin群

Demushkin groups of uncountable rank

本文将Demushkin群理论从可数生成情形推广到任意秩，研究其代数性质、同构类、绝对Galois群实现及投射有限完备化。

Demushkin groups and inverse Galois theory for pro-p-groups of finite rank and maximal p-extensions

This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p)_{E} \ge 3$, for some prime number $p$, then $\mathcal{G}(E(p)/E)$ does not possess nontrivial proper decomposition groups. When $r(p)_{E} = 2$, it describes the decomposition groups of $\mathcal{G}(E(p)/E)$. The paper shows that if $(K, v)$ is a $p$-Henselian valued field with $r(p)_{K} \in \mathbb N$ and a residue field of characteristic $p$, then $P \cong \widetilde P$ or $P$ is presentable as a semidirect product $\mathbb …

Counting Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions using Massey products

We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb U}_4({\mathbb F}_p)$ consisting of unipotent four by four matrices over ${\mathbb F}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condit…

Relations in the maximal pro-$p$ quotients of absolute Galois groups

We observe that some basic but fundamental constructions in Galois theory can be used to obtain some interesting restrictions on the structure of Galois groups of maximal $p$-extensions of fields containing a primitive $p$th root of unity. This is an extension of some significant ideas of Demushkin, Labute and Serre from local fields to all fields containing a primitive $p$th root of unity. Our techniques use certain natural simple Galois extensions together with some considerations in Galois cohomology and Massey products.

The Kernel Unipotent Conjecture and the vanishing of Massey products for odd rigid fields

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey Conjecture for $n\geq 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of $p…