The algebraic closure of a formal power series field
This paper is devoted to the study of the algebraic closure \(\overline {k((T) )}\) of the field of meromorphic formal power series \(k((T))\) in one indeterminate over a field \(k\) of characteristic \(p \neq 0\). It is proved that this algebraic closure is the limit of an increasing sequence of Galois extensions starting from the Puiseux field \(k((T^{{1 \over \infty}})) = \cup_{n = 1}^\infty k((T^{{1 \over n}}))\). The separable closure of \(k((T))\) is also described. We note that the paper contains several interesting examples. Some of them are based on techniques developed by \textit{F. J. Rayner} [Glasg. Math. J. 9, 146-151 (1968; Zbl 0159.05302)], \textit{P. Ribenboim} and \textit{L. Van der Vries} [Can. Math. Bull. 27, 313-315 (1984; Zbl 0548.12013)] and \textit{D. Ştefănescu} [Bull. Math. Soc. Sci. Math. Repub. Soc. Roum., Nouv. Ser. 26, 77-91 (1982; Zbl 0487.12015), ibid. 27, 169-178 (1983; Zbl 0515.12018)]. The author proves that there exist two generalized power series with the same support but such that one is algebraic over \(k((T))\) and the other is transcendental.
scientific article; zbMATH DE number 871696
scientific article