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[tex]\mathbb{C}((t)) = \{t^{-n_0}\sum_{i=0}^{\infty}a_it^i\ :\ n_0 \in \mathbb{N}, a_i \in \mathbb{C}\}[/tex]

What is its algebraic closure? My notes say that it is "close" to:

[tex]\bigcup _{m \in \mathbb{N}}\mathbb{C}((t))(t^{1/m})[/tex]

where [itex]\mathbb{C}((t))(t^{1/m})[/itex] is the extention of the field of Laurent series by the element t^{1/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 overC((t)) has a root in X, but how do I show that there is no intermediate field between X andC((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?

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# Algebraic Closure of Laurent Series

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