Some remarks on orderings under finite field extensions (Q1183110)

Let \(X_ K\) be the space of orderings of a field \(K\). The image of the restriction mapping \(r_{L/K}:X_ L\to X_ K\), where \(L/K\) is a finitely generated field extension, is clopen in \(X_ K\). Moreover, every clopen subset \(U\) of \(X_ K\) is of the form \(r_{L/K}(X_ L)\), for some finitely generated extension \(L/K\) [\textit{R. Elman}, \textit{T. Y. Lam} and \textit{A. R. Wadsworth}, J. Reine Angew. Math. 306, 7-27 (1979; Zbl 0398.12019)]. One can ask whether it is possible to find a finite extension \(L/K\) that realizes a given clopen subset \(U\) of \(X_ K\). \textit{A. Prestel} [Rocky Mt. J. Math. 19, 897-911 (1989; Zbl 0702.11021)] showed that it is possible when \(K\) is a Hilbertian field. In the first part of this paper the author discusses the Hilbertian case from the point of view of possible degree \([L:K]\). He obtains an upper bound for \([L:K]\) which depends on the presentation of \(U\) with the help of Harrison subbasis. In the second part the results are extended to some larger class of fields. However, the bound obtained in this case is weaker than in the Hilbertian case.