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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Journal of Singularities, Worldwide Center of Mathematics, LLC, 2107, <10.5427/jsing.2017.16a>
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https://hal.archives-ouvertes.fr/hal-01586836
Contributeur : Guillaume Rond <>
Soumis le : mercredi 13 septembre 2017 - 12:26:37
Dernière modification le : jeudi 14 septembre 2017 - 01:05:49

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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, 2107, <10.5427/jsing.2017.16a>. <hal-01586836>

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