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On finite embedding problems with abelian kernels. (English) Zbl 1489.12016

The inverse Galois problem over a field \(K\) asks whether every finite group is a Galois group over \(K\). A generalization of this problem is given by the finite embedding problem. More precisely, a finite embedding problem over \(K\) is an epimorphism \(\alpha:G \rightarrow \mathrm{Gal}(L/K)\), where \(G\) is a finite group and \(L/K\) a Galois extension. We say that \(\alpha\) splits if there is a morphism \(\tau: \mathrm{Gal}(L/K) \rightarrow G\) such that \(\alpha \circ \tau =\mathrm{Id}_{\mathrm{Gal}(L/K)}\). A solution to \(\alpha\) is an isomorphism \(\beta: \mathrm{Gal}(F/K) \rightarrow G\), where \(F/K\) is a Galois extension of \(L/K\), such that \(\alpha \circ \beta\) is the restriction.
It is known (see [M. D. Fried and M. Jarden, Field arithmetic. 3rd revised ed. Berlin: Springer (2008; Zbl 1145.12001)]) that: given a Hilbertian field \(K\), every finite split embedding problem \(\alpha : G \rightarrow \mathrm{Gal}(L/K)\) over \(K\) with abelian kernel has a solution. In this work, the author strengthens this result:
Theorem 1. Let \(K\) be a Hilbertian field, \(\mathcal{S}\) a finite set of Krull valuations of \(K\), and \(\alpha: G \rightarrow\mathrm{Gal}(L/K)\) a finite split embedding problem with abelian kernel over \(K\). Then \(\alpha\) has a solution \(\mathrm{Gal}(F/K)\rightarrow G\) such that \(F \subset L K_v^h\) for every \(v \in \mathcal{S}\), where \(K_v^h\) denotes the Henselization of \(K\) at \(v\).
Two applications of the above theorem are given: to the Beckmann-Black problem and inverse Galois theory over division rings. In fact, the author proves
Theorem 2. Let \(G\) be a non-trivial finite solvable group, \(K\) an arbitrary field, and \(F/K\) a Galois extension of group \(G\). There exist \(t_0 \in K\) and a Galois extension \(E/K(T)\) of group \(G\) with \(E\not\subset\overline{K}(T)\) such that the specialization \(E_{t_0} /K\) of \(E/K(T)\) at \(t_0\) equals \(F/K\).
and
Theorem 3. Let \(H\) be a division ring and \(\sigma\) an automorphism of \(H\) of finite order. Every finite semiabelian group is a Galois group over \(H(T, \sigma)\), where \(H(T,\sigma)\) denotes the skew field of rational fractions associated with \(H\) and \(\sigma\).

MSC:

12E25 Hilbertian fields; Hilbert’s irreducibility theorem
12F12 Inverse Galois theory
12F10 Separable extensions, Galois theory

Citations:

Zbl 1145.12001
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References:

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