Generalized Puiseux expansions and their Galois groups
The author considers an algebraically closed field \(k\) of characteristic \(p\) and the field \(k((X^{\mathbb Q,p}))\) of the restricted power series over \(k\) [see \textit{P. Ribenboim}, Manuscr. Math. 75, 115-150 (1992; Zbl 0767.12001)]. He obtains criteria for special restricted power series to be algebraic over the field of meromorphic formal power series \(k((T))\). It is proved that if \(k\) is not the algebraic closure of its prime field, \(Y\in k\) is transcendental over the prime field of \(k\), \(L\) is a finite field contained in \(k\), \(f_i(Y)\in L[Y]\) and there exists a positive integer \(M\) such that \(\deg f_i(Y)\leq M\) for all \(i\), then the element \(f=\sum_{i=1}^\infty f_i(Y)X^{-1/p^i}\) is algebraic over \(k((X))\) if and only if it is periodical. Finally, the author computes the Galois groups of some particular restricted power series.
scientific article; zbMATH DE number 1029006