Allowable groups and cogalois extensions (Q1903704)

Let \(K/F\) be an extension of number fields. The torsion of \(K/F\) is defined by \({\mathcal T} (K/F) : = \{a \in K \mid a^m \in F^*\) for some \(m \in \mathbb{N}\}\). Then \(\text{cog} (K/F) : = {\mathcal T} (K/F)/F^* \subseteq K^*/F^*\) is called the `cogalois group of \(K/F\)'. \textit{C. Greither} and \textit{D. K. Harrison} [J. Pure Appl. Algebra 43, 257-270 (1986; Zbl 0607.12015)] introduced the concept of cogalois extensions. \(K/F\) is called `cogalois' if \(|\text{cog} (K/F) |\leq [K : F]\) and \(K = F({\mathcal T} (K/F))\). Let \(r\) be an integer and let \(m\) be a divisor of \(r\) such that each prime divisor of \(r\) divides \(m\) and \(4 |m\) if \(4 |r\). Set \(D_{r,m} : = \{x \bmod r \in (\mathbb{Z}/r \mathbb{Z})^* \mid x \equiv 1 \bmod m\}\). We call a group \(G\) `allowable' if there is an exact sequence \(1 \to H \to G \to D_{r,m} \to 1\), with \(H\) an abelian group, the exponent of \(G\) divides \(r\) and \(D_{r,m}\) acts on \(H\) by scalar multiplication. The paper under review gives a necessary and sufficient condition on the field \(F\) in order to have a Galois and cogalois extension \(K/F\) with Galois group an arbitrary allowable group \(G\). This condition depends on the structure of the abelian group \(G/G'\) \((G' =\) commutator subgroup of \(G)\) and certain roots of unity. An allowable group is nilpotent, hence a direct product of its Sylow groups, which are also allowable. From this we get a reduction to the case of allowable \(p\)-groups.