The algebraic closure of the power series field in positive characteristic (Q2750838)

There is constructed an algebraic closure of the field \(K((t))\) of meromorphic power series over a field of positive characteristic \(p\). The author introduces the class of twist--recurrent power series over \(K\) as the family of general power series with the support in the set NEWLINE\[NEWLINE S_{a,b,c} = \Bigl\{\tfrac 1a (n-b_1p^{-1}-b_2p^{-2}-\cdots) : n\geq -b,\;b_i\in\{0, \dots, p-1\},\;\sum b_i \leq c\Bigr\}, NEWLINE\]NEWLINE where \(a\in\mathbb{N}\), \(b,c\geq 0\) and that satisfy to conditions given by twist-recurrent functions. This is a subset of the restricted power series considered by other authors [cf., for example, \textit{P. Ribenboim}, Manuscr. Math. 75, 115-150 (1992; Zbl 0767.12001)]. There is given a representation for twist-recurrent series supported on \(S_{1,b,c}\) and it is proved that the algebraic closure of \(K((t))\) is given by the twist-recurrent power series. The paper also contains several applications. We mention a description of algebraic elements over \(L((t))\) in the case \(L\) is perfect and the study of periodic series.