The algebraic closure of the power series field in positive characteristic

Abstract:

For $K$ an algebraically closed field, let $K((t))$ denote the quotient field of the power series ring over $K$. The “Newton-Puiseux theorem” states that if $K$ has characteristic 0, the algebraic closure of $K((t))$ is the union of the fields $K((t^{1/n}))$ over $n \in \mathbb {N}$. We answer a question of Abhyankar by constructing an algebraic closure of $K((t))$ for any field $K$ of positive characteristic explicitly in terms of certain generalized power series.