The embedding problem with non-abelian kernel for local fields
type: Item
instance of: scholarly article
title: The embedding problem with non-abelian kernel for local fields
zbMATH Open document ID: 1208.11127
author: V. V. Ishkhanov
author: B. B. Lurie
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-01-19T00:00:00Z
review text: Let \(k\) be a finite extension of the field \(\mathbb Q _ p\) of \(p\)-adic numbers of degree \(d\), \(\overline F\) the Galois group of the maximal \(p\)-extension \(k(p)/K\), and let \(K/k\) be a \(p\)-extension with Galois group \(F\). Consider the exact sequence \(1 \to B \to G \to F \to 1\) of \(p\)-groups and denote by \(\varphi \) the homomorphism \(\varphi \) of \(G\) onto \(F\) from this sequence. The embedding problem consists of finding a Galois extension (or at least a Galois algebra) \(L\) of \(k\) with a Galois group \(G\), such that \(K \subseteq L\) and the restriction of automorphisms \(g \in G\) upon \(K\) coincides with \(\varphi (g)\). The solution is obtained easily in case \(k\) does not contain a primitive \(p\)-th root of unity, since then \(\overline F\) is a free pro-\(p\)-group of rank \(d + 1\), by a theorem of Shafarevich. When the kernel \(B\) of this problem is abelian, it was shown by \textit{S. P. Demushkin} and \textit{I. R. Shafarevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 23, 823--840 (1959; Zbl 0093.04404)] that the problem is solvable if and only if the Faddeev-Hasse compatibility condition is fulfilled; for the answer over an arbitrary ground field, found by \textit{A. Yakovlev} [Izv. Akad. Nauk SSSR, Ser. Mat. 28, 645--660 (1964; Zbl 0126.27402)]. Henceforth, we assume that \(k\) contains a primitive \(p\)-th root of unity, \(B\) is nonabelian and \(B'\) is the derived subgroup of \(B\), and for each pro-\(p\)-group \(P\), we denote by \(d(P)\) the cardinality of any minimal system of topological generators of \(P\). The paper under review deals with the question of whether the stated embedding problem is solvable if and only if the associated problem corresponding to the exact sequence of groups \(1 \to B/B' \to G/B' \to F \to 1\), is solvable. It shows that the answer is affirmative in the following two cases: (i) \(d(\overline F) \geq d(F) + 3\); (ii) \(d(\overline F) \geq d(F) + 2\) and \(K\) contains a primitive root of unit…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 5660150
zbMATH Keywords: embedding problem
zbMATH Keywords: Faddeev-Hasse compatibility condition
zbMATH Keywords: local field
zbMATH Keywords: associated embedding problem with abelian kernel
MaRDI profile type: Publication
full work available at URL: s10958-009-9588-7
OpenAlex ID: W2059803311
cites work: Q3269219
cites work: Q5333434
cites work: Q5619167
cites work: Q4346209
DOI: 10.1007/S10958-009-9588-7
description: scientific article; zbMATH DE number 5660150