# zbMATH — the first resource for mathematics

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 $$\overline{\mathbb{Q}}_ p)$$.

##### MSC:
 12J10 Valued fields