Completions of r.a.t.-valued fields of rational functions
Let \(\mathbb{Q}_ p\) denote the field of \(p\)-adic numbers, \(\Omega\) its algebraic closure, and \(\overline{\Omega}\) the completion of \(\Omega\) with respect to the unique extension of the \(p\)-adic valuation \(v\) at \(\Omega\). This paper deals with the structure of complete fields \(L\), \(\mathbb{Q}_ p \subseteq L \subseteq \overline{\Omega}\). Let \(F_ c(\overline{\Omega}/\mathbb{Q}_ p)\) denote the set of all complete intermediate fields of \(\overline{\Omega}/\mathbb{Q}_ p\). If \([L:\mathbb{Q}_ p]\) is finite, there exists \(a \in L\) such that \(L = \mathbb{Q}_ p(a)\). The authors show in this paper that if \([L : \mathbb{Q}_ p]\) is infinite, then there exists \(z \in L\) (\(z\) transcendental over \(\mathbb{Q}_ p\)) such that \(L\) is the completion of \(\mathbb{Q}_ p(z)\) with respect to the restriction of the valuation \(v\) of \(\overline{\Omega}\) to \(\mathbb{Q}_ p(z)\). They use the result that the correspondences \(F(\Omega/K) \supset l \to \overline{l} \in F_ c(\overline{\Omega}/K)\) and \(F_ c(\overline{\Omega}/K) \supset L \to L \cap \Omega \in F(\Omega/K)\) are one-to-one and inverse to each other. The authors also prove that \(L\) is the completion of \(Q(x)\) with respect to a valuation \(w\), which extends \(\nu\) to \(Q(x)\) where \(x\) is transcendental over \(Q\). The valuation \(w\) is the residual algebraic torsion extension of \(v\).
scientific article; zbMATH DE number 729071