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Power series and \(p\)-adic algebraic closures. (English) Zbl 0980.12002

Let \(K\) be an algebraically closed field of characteristic \(p>0\), \(W(K)\) the corresponding Witt ring. The author gives an explicit description of the \(p\)-adic completion of the integral closure \(\overline{W(K)}\) of \(W(K)\) in an algebraic closure of its fraction field. The construction uses generalized power series introduced by B. Poonen [Enseign. Math., II. Sér. 39, 87-106 (1993; Zbl 0807.12006)], and the description of the algebraic closure of a field of power series given by the author [K. S. Kedlaya, Proc. Am. Math. Soc. 129, 3461-3470 (2001; Zbl 1012.12007)]. The basic step of an independent interest is a construction of a surjection of \(W(K[[t]])\) onto the \(p\)-adic closure of \(\overline{W(K)}\).

MSC:

12J25 Non-Archimedean valued fields
11S85 Other nonanalytic theory
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