## 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.

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 $\omega$.