The embedding problem with non-abelian kernel for local fields
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 unity of degree equal to the exponent of \(B\). Here we recall that, by an earlier result of Lur'e (published in 1973), the answer is also affirmative in the case of \(d(G) = d(F)\).
associated embedding problem with abelian kernel
scientific article; zbMATH DE number 5660150