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

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.
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Submitted on : Monday, May 9, 2016 - 3:18:42 PM
Last modification on : Monday, March 4, 2019 - 2:04:19 PM

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Guillaume Rond. About the algebraic closure of the field of power series in several variables in characteristic zero. Journal of Singularities, Worldwide Center of Mathematics, LLC, 2017, 16, pp.1-51. ⟨10.5427/jsing.2017.16a⟩. ⟨hal-01313076⟩

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