Valuation fans and residually closed Henselian fields
A field \(R\) is called a HRC-field if \(R\) admits a henselian valuation \(v\) with real closed residue field. Due to \textit{N. Schwartz} [Séminaire sur les Structures Algébriques Ordonnées, Vol.1, 67-78, Publications de l'Université Paris 7 32, 65-78 (1990; Zbl 0713.12003)] one knows that \(R\) is a HRC- field if and only if \(R\) is real closed with respect to a signature of \(R.\) A preordering \(T\) of a field \(K\) is called a valution fan of \(K\) if for any \(x\not\in\pm T\) we have \(1\pm x\in T\) or \(1\pm x^{-1}\in T.\) Since for a valuation fan \(T\subset K\) the canonical projection \( f_T: K^*\rightarrow K^*/T^* \) is a signature of \(K\), there is a close relationship between HRC-fields and valuation fans. The authors start the paper with collecting basic results on valuation fans and then they concentrate on valuation fans of HRC-fields. Among others they prove that \(R\) is a HRC-field if and only if \(R\) is a hereditarily pythagorean field and for any natural number \(n, \) the subset \( R^{2n}\) is a valuation fan. Thus due to the result of \textit{B. Jacob} [Pac. J. Math. 93, 95-105 (1981; Zbl 0485.12016)] on hereditarily pythagorean fields they get that the class of HRC- fields is an elementary class. \textit{R. Brown}, \textit{T. Craven} and \textit{M. Pelling} [Ill. J. Math. 30, 66-78 (1980; Zbl 0559.12013)] described all ordered fields satisfying Rolle's theorem for all polynomials. They turned out to be HRC-fields with an extra assumption on the value group. The authors of the paper under review examine HRC-fields from of point of view of the validity of Rolle's theorem for polynomials of some degrees. A field extension \(R\) of the field \(K\) is called an algebraic HRC-extension of \(K\) if \(R\) is a HRC-field which is algebraic over \(K\). It turns out that two algebraic HRC-extensions of a fixed field \(K\) has not to be \(K\)-isomorphic. However, the authors prove that two algebraic HRC-extensions of \(K\) are \(K\)-isomorphic if they induce on \(K\) the same chain of valuation fans. Thus an algebraic HRC-extension of \(K\) can be considered as the real closure of \(K\) with respect to a chain of valuation fans of \(K\).
scientific article; zbMATH DE number 1309129
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