Algebraic Closure of Laurent Series

by AKG
Tags: algebraic, closure, laurent, series
 Sci Advisor HW Helper P: 2,586 Define: $$\mathbb{C}((t)) = \{t^{-n_0}\sum_{i=0}^{\infty}a_it^i\ :\ n_0 \in \mathbb{N}, a_i \in \mathbb{C}\}$$ What is its algebraic closure? My notes say that it is "close" to: $$\bigcup _{m \in \mathbb{N}}\mathbb{C}((t))(t^{1/m})$$ where $\mathbb{C}((t))(t^{1/m})$ is the extention of the field of Laurent series by the element t1/m. Is this in fact the closure? If not, what is it? Also, how would I prove that something is the algebraic closure of this field? I mean, if X is the algebraic closure, then one thing is to prove that every polynomial over C((t)) has a root in X, but how do I show that there is no intermediate field between X and C((t)) that also has this property? I.e. it's one thing to show that a field X contains the algebraic closure of another field Y, but how do I show that it IS the algebraic closure of Y?