Arithmetic lifting of dihedral extensions
The standard and usual approach to inverse Galois problems relies on the construction of Galois branched coverings of the projective line and on their specialization using the Hilbert Irreducibility Theorem. Therefore it is fundamental to know whether every Galois extension of number fields is the specialization of a branched covering of the projective line with the same Galois group. The author proves that this holds true for every Galois extension of number fields whose Galois group is a dihedral group of odd degree (\(D_n\), with \(n\) odd). The same positive result was obtained by \textit{S. Beckmann} when the Galois group is an abelian finite group [J. Algebra 165, 430-451 (1994; Zbl 0802.12003)]. The author reinterprets the work of Beckmann in terms of étale cohomology and then generalizes to the case of dihedral groups of odd degree.
scientific article; zbMATH DE number 1203903
scientific article