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

Journal of Singularities
volume 16 (2017), 1-51

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

DOI: 10.5427/jsing.2017.16a

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Abstract:

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
CNRS
Centrale Marseille, I2M
Marseille, France
email: guillaume.rond@univ-amu.fr