Puiseux’s theorem

March 22, 2011

In Local Fields, Serre uses ramification groups to prove that the algebraic closure of the power series field K((T)) over an algebraically closed field K of characteristic zero is the colimit of K((T^{1/n})) for n > 0; 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 K. 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.


2 Responses to “Puiseux’s theorem”

  1. Scott Carnahan Says:

    You probably want K to have characteristic zero. Otherwise, you run into problems with polynomials like x^p - x - 1/T. In positive characteristic, the order type of the set of exponents with nonzero coefficients can be as large as \omega^\omega, while the Puiseux field only supports \omega.

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