Galois groups of order \(2n\) that contain a cyclic subgroup of order \(n\).
Let \(n\geq 2\) be an integer. Let \(F\) be a field with \(\text{char\,}F\nmid n\) and let \(L/F\) be a Galois quadratic extension (that is, if \(\text{char\,}F = 2\nmid n\), \(L/F\) is a separable extension). Assume \(L\) contains a primitive \(n\)-th root of unity \(\zeta\). For a cyclic Galois extension \(M = L(\alpha^{1/n})/L^\times)\) of degree \(n\) such that \(M\) is Galois over \(F\), the authors determine, in terms of the action of \(\text{Gal}(L/F)\) on \(\alpha\) and \(\zeta\), what group occurs as \(\text{Gal}(M/F)\). To do this, they first classify the possible groups that can arise as \(\text{Gal}(M/F)\). They count the number of isomorphism classes of groups of order \(2n\) containing a cyclic subgroup of order \(n\). Let \(\text{Gal}(L/F) = \{1,\sigma\}\). One checks that \(M\) is Galois over \(F\) if and only if \(\sigma(\alpha)=\alpha^t \beta^n\) for some \(\beta\in L^\times\), with \(t^2\equiv 1\bmod n\). The authors give a general description of the group \(\text{Gal}(M/F)\) in terms of \(\alpha,\beta\) and \(t\). In order to make the determination of \(\text{Gal}(M/F)\) more explicit, they obtain precise descriptions of the \(\alpha\) satisfying equalities \(\sigma(\alpha) =\alpha^t\beta^n\). This allows them, in the two final sections, to pin down in detail the circumstances under which a given group arises.
scientific article; zbMATH DE number 2101134