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Galois realization of Schur covers of dihedral groups of \(2\)-power order. (English) Zbl 1514.12001

In this paper, the three authors study automatic realizations of the dihedral, semi-dihedral and generalized quaternion 2-groups as Galois groups. Recall that if any field admitting a \(G\)-extension also admits an \(H\)-extension, then we write \(G\Rightarrow H\) and call this statement an automatic realization. In the main result of this paper Theorem 1.1 it is shown we have the automatic realization \(G\Rightarrow H\) for any choice of \(G\) and \(H\) among the mentioned above groups, with one additional condition on the existence of a certain \(C_4\) extension over the base field. The authors use the isoclinism of these groups to illustrate the Galois correspondence related to the \(C_4\) extensions in Theorem 3.1.

MSC:

12F12 Inverse Galois theory
11R32 Galois theory

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References:

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