2011/03/10 by Ivan D. Chipchakov, Chipchakov, I. D.
Mathematics · #12J10 12F10 12E30 #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1103.2114
openalex publication_date 2011/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper proves that if E is a field, such that the Galois group G(E(p)/E) of the maximal p-extension E(p)/E is a Demushkin group of finite rank r(p)E ≥ 3, for some prime number p, then G(E(p)/E) does not possess nontrivial proper decomposition groups. When r(p)E = 2, it describes the decomposition groups of G(E(p)/E). The paper shows that if (K, v) is a p-Henselian valued field with r(p)K ∈ \mathbb N and a residue field of characteristic p, then P ≅ \widetilde P or P is presentable as a semidirect product \mathbb Zpτ \rtimes \widetilde P, for some τ∈ \mathbb N, where \widetilde P is a Demushkin group of rank ≥ 3 or a free pro-p-group. It also proves that when \widetilde P is of the former type, it is continuously isomorphic to G(K ′(p)/K ′), for some local field K ′ containing a primitive p-th root of unity.