On the structure of nonarchimedean exponential fields. I
Given an ordered field \(K\), we are interested in the structure of its ordered additive group \((K,+, 0,<)\) and that of its ordered multiplicative group \((K^{>0}, \cdot, 1,>)\). To that end, we consider those groups as valued modules, for their natural (Krull) valuations. In section 2, we give a survey of the general theory on valued and ordered modules that we need. In section 3, we give lexicographic decompositions of \((K,+, 0,<)\), and of \((K^{>0}, \cdot, 1,<)\). We then proceed to compute the natural valuation and skeleton of \((K^{>0}, \cdot, 1,<)\) in terms of those of \((K,+, 0,<)\). We use this computation to provide necessary and sufficient conditions on the value group \(v(K)\) and residue field \(\overline {K}\), for the \(L_{\infty\omega}\)-equivalence of the above mentioned groups. In particular, we show that if \(K\) is root closed for positive elements, then its additive group of infinitesimals \(I_ v\) is always infinitary equivalent to its multiplicative group of 1-units \(1+I_ v\). We conclude that if \(K\) is countable, or if \(K\) is a power series field, then \(i_ v\) and \(I+I_ v\) are always isomorphic. We then apply the results to exponential fields. We describe \(v(K)\) in that case, and introduce the notion of an exponential group. Finally, if \(K\) is countable or a power series field, we derive necessary and sufficient conditions on \(v(K)\) and \(\overline {K}\) for \(K\) to be exponential. In the countable case, we get a structure theorem for \(v(K)\) and derive a construction method for all countable nonarchimedean exponential fields.
countable nonarchimedean exponential fields
scientific article; zbMATH DE number 769482
scientific article