About the algebraic closure of the field of power series in several variables in characteristic zero

Guillaume Rond

Journal of Singularities
volume 16 (2016), 1-51

Received: 3 March 2016. Received in revised form: 2 February 2017.

DOI: 10.5427/jsing.2017.16a

Add a reference to this article to your citeulike library.


We begin this paper by constructing different algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and is constructed via a generalization of the Newton-Puiseux method for this valuation.

Then we study the Galois group of a polynomial with power series coefficients. In particular by examining more carefully the case of monomial valuations we are able to give several results concerning the Galois group of a polynomial whose discriminant is a weighted homogeneous polynomial times a unit. One of our main results is a generalization of Abhyankar-Jung Theorem for such polynomials, classical Abhyankar-Jung Theorem being devoted to polynomials whose discriminant is a monomial times a unit.

Mathematical Subject Classification (2000):

Primary: 13F25. Secondary: 11J25, 12J20, 12F99, 13J05, 14B05, 32B10.

Author(s) information:

Guillaume Rond
Aix-Marseille Université
Centrale Marseille, I2M
Marseille, France
email: guillaume.rond@univ-amu.fr