In Local Fields, Serre uses ramification groups to prove that the algebraic closure of the power series field over an algebraically closed field of characteristic zero is the colimit of for ; this is, as he calls it, a formal analog of the usual Puiseux theorem. If I am not being silly, there is an easier way to see this. For any finite extension, the unique valuation on the power series field extends uniquely (since these are all complete); the residue class field extension is trivial, since that of the power series field is already . Thus the extension is tamely ramified, and now by general facts, any totally and tamely ramified extension of a local field is obtained by adjoining a root of a uniformizer. (Cf. Lang’s Algebraic Number Theory; this follows from Hensel’s lemma.)
I added this to the chapter on completions. Also, as promised in the comments to this post, there is now some material on finite presentation in a chapter currently loosely marked various. It is far from complete, though.