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Suppose that $k$ is an arbitrary field. Consider the field$k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$of formal power series in the variables $x_1,...,x_n$, with coefficients in$k$. Suppose that $\sigma$ is a formal power series in $x_1,...,x_n$ withcoefficints in the algebraic closure of $k$. We give a very simple necessaryand sufficient condition for $\sigma$ to be algebraic over $k((x_1,...,x_n))$.  As an application of our methods, we give a characterization of valuationrings $V$ which dominate an excellent, Noetherian local domain $R$ of dimensiontwo, and such that the rank increases after passing to the completion of abirational extension of $R$.

Algebraic series and valuation rings over nonclosed fields