\(p\)-Henselian fields
This paper studies a special instance of a more general concept introduced by L. Bröcker: Given some normal algebraic field extension \(\Omega/F\), the valued field \((F,v)\) is called \(\Omega\)-henselian if \(v\) extends uniquely to \(\Omega\). Now for a prime \(p\), let \(\Omega = F(p)\) be the maximal Galois-\(p\)-extension of \(F\). \(v\) is \(p\)-henselian if it is \(F(p)\)-henselian. \(p\)-henselian valuations are used in \(K\)-theory, Galois cohomology [e.g. in \textit{A. Wadsworth}, Pac. J. Math. 105, 473-496 (1983; Zbl 0506.12024)], and in studying the Galois group of \(F(p)/F\) [e.g. in \textit{B. Jacob} and \textit{A. Wadsworth}, Trans. Am. Math. Soc. 293, 693-721 (1986; Zbl 0597.16016)] and [\textit{I. Efrat}, Free pro-\(p\) product decomposition of Galois groups, preprint (1993)]. The main results are all proved under the following assumption on \(F\): \(F\) contains a primitive \(p\)-th root of unity \(\zeta_p\), resp. \(\text{char} (F) = p\). For such \(F\), \(F^{(p)}\) denotes the multiplicative group \(F^p\) resp. the additive group \(\{x^p - x; x \in F\}\). The \(p\)-topology \(\tau_p\) on \(F\) is defined as the coarsest topology for which \(F^{(p)}\) is open and for which linear transformations resp. Möbius transformations are continuous. The author shows that if \(F\) is not \(p\)-closed (i.e. \(F \neq F(p))\), then a nontrivial valuation \(v\) on \(F\) admits a nontrivial \(p\)-henselian coarsening if and only if \(v\) induces the \(p\)-topology on \(F\). In this case, a uniformly parametrized base for \(\tau_p\) is provided. As a corollary, he obtains the main result of the paper, which asserts that if \(F\) is not \(p\)-closed, then to admit a nontrivial \(p\)-henselian valuation is an elementary property in the language of fields, unless \(p = 2\) and \(F\) is euclidean. Indeed, since ``to be a \(V\)-topology'' can be elementarily expressed in terms of a base of the topology [cf. \textit{A. Prestel} and \textit{M. Ziegler}, J. Reine Angew. Math. 299/300, 318-341 (1978; Zbl 0367.12014)], the property expressing that \(\tau_p\) is a \(V\)-topology is elementary. Now under the hypothesis of the corollary, a \(V\)-topology arises necessarily from a valuation, hence the assertion of the corollary follows from the first result. Finally, the valuation ring \({\mathcal O}\) of the canonical \(p\)-henselian valuation \(v_p\) of \(F\) is considered: \(v_p\) is defined to be the coarset \(p\)-henselian valuation with \(p\)-closed residue field if there are any such, and to be the finest \(p\)-henselian valuation otherwise. It is shown that \({\mathcal O}\) is elementarily definable unless \(p = 2\) and the residue field is euclidean.
scientific article; zbMATH DE number 784063
scientific article