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International audienceWe begin this paper by constructing different algebraically closedfields containing an algebraic closure of the field of power series in severalvariables over a characteristic zero field. Each of these fields depends on thechoice of an Abhyankar valuation and is constructed via a generalization ofthe 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 weare able to give several results concerning the Galois group of a polynomialwhose discriminant is a weighted homogeneous polynomial times a unit. Oneof our main results is a generalization of Abhyankar-Jung Theorem for suchpolynomials, classical Abhyankar-Jung Theorem being devoted to polynomialswhose discriminant is a monomial times a unit

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