Tame Galois \(p\)-extensions of \(p\)-Henselian fields
A valued field \(K\) is called \(p\)-Henselian if its valuation extends uniquely to the maximal Galois \(p\)-extension \(K(p)\) of \(K\). The main theorem of the paper under review gives a determination of the structure of the Galois group \(G_K(p)\) of the extension \(K(p)/K\) in case the characteristic of the residue field \(\overline{K}\) is different from \(p\). We get \[ G_K(p)\simeq T\rtimes G_{\overline{K}} (p), \] where \(T\) is the inertia group of \(G_K(p)\). Further \(T\simeq \mathbb{Z}_p^\mu\) for some \(\mu\). The operation of \(G_{\overline{K}}(p)\) on \(\mathbb{Z}_p^\mu\) is given by the \(p\)-cyclotomic characters. The possible values of \(\mu\) are described.
scientific article; zbMATH DE number 1334554