Completion of r. t. extensions of local fields. II
Let \((K,v)\) be a local field. An extension \(w\) of \(v\) to \(K(X)\) is called residual transcendental (r.t.-extension) if the residue field of \(w\) is a transcendental extension of the residual field of \(v\). In the first part of this paper [Math. Z. 221, 675-682 (1996; Zbl 0852.12003)] the authors described the completion \((\widetilde{K(X)},\widetilde w)\) of \((K(X),w) \), where \(w\) is the Gauss r.t.-extension of \(v\). In this paper they consider the general case of any r.t.-extension of \(v\). For the description of elements of the completion they use the embedding of \((K(X),w)\) into the valued field \(K' \{\{X-a.\delta\}\}\) of \(\delta\)-formal Laurent series over the local field \((K', v')\), where \(\delta\) is an appropriate rational number, and \(K'=K(a) \subset \overline K\), \(a\in\overline K\) (the algebraic closure of \(K)\). Moreover, they describe \((\widetilde{K(X)},\widetilde w)\) as a finite extension of \((\widetilde {K(r)}, \widetilde{w_0})\), where \(r\in K(X)\) and \(w_0\) is the Gauss r.t.-extension of \(v\).
scientific article; zbMATH DE number 1279128