Tame fields and tame extensions
Let \(V\) be a henselian valuation of any rank of a field \(K\), and let \(\overline V\) be the extension of \(V\) to a fixed algebraic closure \(\overline K\) of \(K\). In this paper, the author proves that \((K,V)\) is a tame field, i.e., every finite extension of \((K,V)\) is tamely ramified, if and only if \(\forall\alpha\in\overline K\setminus K\), there exists \(a\in K\) such that \(\overline V(\alpha-a)\geq\Delta_K(\alpha)\), where \(\Delta_K (\alpha)= \min\{ \overline V(\alpha'-\alpha)|\alpha'\) varies over all \(K\)-conjugates of \(\alpha\}\). In the course of proving the main result, the author shows that if \((K',V')\) is any finite Galois tame extension of a henselian valued field \((K,V)\), then to each \(\alpha\in K'\) there corresponds \(a\in K\) such that \(V'( \alpha-a)\geq\Delta_K(\alpha)\). The author gives an example showing that if \((K', V')\) is a finite Galois extension of a henselian field \((K,V)\), then it is not necessarily the case that the statement to each \(\alpha\in K'\) there corresponds \(a\in K\) such that \(V'(\alpha-a)\geq\Delta_K(\alpha)\) implies the statement \((K', V')\) is a tame extension of \((K,V)\). However the author shows that the implication holds under the added assumption that \(\Delta_K(\alpha)= \omega_K (\alpha)\) for some generator \(\alpha\) of \(K'/K\), where \(\omega_K(\alpha)=\infty\) if \(\alpha\in K\) and \(\omega_K(\alpha)=\max\{\overline V(\alpha-\alpha')\mid \alpha' \neq \alpha\) varies over all \(K\)-conjugates of \(\alpha\}\) otherwise.
scientific article; zbMATH DE number 1214118