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Maximally complete fields. (English) Zbl 0807.12006
I. Kaplansky [Duke Math. J. 9, 303-321 (1942; Zbl 0061.055)] proved that among all valued fields with a given divisible value group \(G\), a given algebraically closed residue field \(R\) and a given restriction to the minimal subfield, there is a maximal one such that every other can be embedded in it. In the present paper the author gives a construction of such a valued field. He introduces at first a \(p\)-adic analogy of the Malcev-Neumann ring (i.e. a ring \(R((G))= \{\alpha= \sum_{g\in G} \alpha_ g t^ g\): \(\alpha_ g\in R\), \(\text{supp} (\alpha)\) is well- ordered in \(G\}\), where \(R\) is a ring and \(G\) is a divisible ordered group) as a factor ring \(A((G))/N\) (for some special valuation ring \(A\)), and he shows that it is a valued field with value group \(G\) and the residue field \(R\) (Section 4). In Section 5 he then proves that the Malcev-Neumann ring (or its \(p\)-adic analogue) is the maximal valued field from the Kaplansky theorem. A generalized construction of the Malcev-Neumann ring is then used for applications (e.g. to the problem of “glueing” two valued fields and to the investigation of the maximally complete immediate extension of \(\overline{\mathbb{Q}}_ p)\).

12J10 Valued fields