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Immediate and purely wild extensions of valued fields. (English) Zbl 0593.12018

In this well written paper the authors present a method for proving Kaplansky’s theorem concerning uniqueness of maximal immediate extensions of a valued field \(K\). Their principal tools and methods are based on an investigation of the structure of the Galois group over \(K\) and its subgroups defined by Hilbert’s ramification theory. Moreover, they utilize some analogy between maximal immediate extension and a completion to use a model theoretic point of view for their construction, especially in the case of simple transcendental immediate extensions of \(K\). At the end of the paper the authors partially define conditions for \(K\) to possess a unique maximal immediate extension.

MSC:

12J10 Valued fields
12L12 Model theory of fields
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References:

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