\(n\)-real valuations and the higher level version of the Krull-Baer theorem.
The authors are concerned with the study of orderings of different levels on skew fields. After an example with an ordering of a given level but no higher ones, they consider a skew field \(D\) with an ordering of level \(n\) and an invariant valuation \(v\). This valuation is called \(n\)-real if the residue class field \(k_v\) has an ordering (of level 1) such that positive elements of the valuation ring are mapped by \(v\) to positive elements of \(k_v\). They show how for a given ordering of level 1 on \(k_v\) a preordering of level \(n\) on \(D\) can be constructed inducing the given ordering on \(k_v\). Next define for any group \(G\) its dual \(G^*=\Hom(G,\mu)\) where \(\mu\) is the set of all complex roots of 1 and call \(\chi\in(D^\times)^*\) a signature of level \(n\) if \(\{0\}\cup\ker\chi\) is a level \(n\) ordering of \(D\). The character \(\chi\) is compatible with \(v\), \(\chi\sim v\), if \(1+{\mathfrak m}_v\subseteq\ker\chi\), where \({\mathfrak m}_v\) is the maximal ideal of \(v\). Such \(\chi\) yields a character \(i^*(\chi)\) of \(k_v\) called the pushdown of \(\chi\) with respect to \(v\). A signature \(\pi\) whose kernel contains all positive elements of the valuation ring is said to be \(n\)-compatible with \(v\). If \(D\) has a valuation \(v\) and \(\pi\) is a signature of level \(n\) of \(k_v\) which is \(n\)-compatible with \(v\), then \(\pi\) is shown to have a pullback of level \(n\) with respect to \(v\) and it is shown how to construct all pullbacks from a given one. The set of all signatures of level \(n\) of \(D\) is written \(\text{Sgn}^n(D)\) and a procedure is described for constructing its elements by means of the \(n\)-real valuation rings on \(D\) whose residue-class field has an Archimedean ordering \(n\)-compatible with \(v\). This leads to a short proof of the classification of all signatures on the ring of quantum polynomials [see \textit{J. Cimprič}, Commun. Algebra 28, No. 1, 103--119 (2000; Zbl 1031.12006)]. The valuation theory so developed is then used to study sums of (permuted) \(n\)-th powers in skew fields and illustrate the results on examples. They show that a real place is not necessarily order compatible, thus answering a question raised by \textit{M. Marshall} and \textit{Y. Zhang} [J. Algebra 212, No. 1, 190--207 (1999; Zbl 0934.16041)].
scientific article; zbMATH DE number 2117274