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.

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This entry was posted on March 22, 2011 at 2:37 am and is filed under announcements.

Tags: finite presentation, Puiseux theorem

July 6, 2012 at 10:51 am

You probably want to have characteristic zero. Otherwise, you run into problems with polynomials like . In positive characteristic, the order type of the set of exponents with nonzero coefficients can be as large as , while the Puiseux field only supports .

July 6, 2012 at 11:27 am

Thanks for the correction — I’ve fixed it.