symmetric monoidal (∞,1)-category of spectra
where is a positive integer, is an integer, and each belongs to . Somewhat more abstractly but also more meaningfully, if is the poset of positive integers ordered by divisibility (so that is the least element), then the field of Puiseux series is the filtered colimit of the diagram of fields
Puiseux series were in essence considered by Isaac Newton, who developed a method of expanding algebraic functions as Puiseux series, based on an analogue of Newton's method of approximating roots. Here is a sample theorem:
It is enough to show that every degree extension of the field of Laurent series is of the form . For this, it suffices that the integral closure of in be of the form .
Generally speaking, let be a complete DVR (discrete valuation ring) with maximal ideal and residue class field , and let be its field of fractions. Let be a degree extension of , and let be the integral closure of in . Then is also a complete DVR. We may write the ideal of as where is the ramification index, and we have
where the last equation holds because as -modules and therefore also as -modules. In the case where , we have that since is algebraically closed, and therefore . In other words, , so we can write where generates the maximal ideal of and is a unit of .
The residue class has an root (again by algebraic closure); in fact a simple root since . By Hensel's lemma?, this lifts to an root of in . The element is thus an root of , and is a generator of the maximal ideal of . Writing this element as , the ring is an -submodule of full rank and integrally closed (being abstractly isomorphic to , which is integrally closed because it’s a principal ideal domain and therefore a unique factorization domain), so that , as was to be shown.
(Intend to solve for in as a Puiseux series in .)
The sketched proof of theorem 1 was extracted from notes on a seminar by Boyarchenko on local class field theory:
and the reader may refer to the classic text by Serre for a fuller treatment:
For a noncommutative generalization see
We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking. But we also show that there exist real radical ideals that do not admit a tropical basis. As an application, we show how to compute the set of singular points of a real tropical hypersurface. i.e. we compute the real tropical discriminant.