Article
Rank one discrete valuations of power series fields
Author/s | Herrera Govantes, Francisco Javier
Olalla Acosta, Miguel Ángel ![]() ![]() ![]() ![]() ![]() ![]() Vicente Córdoba, José Luis |
Department | Universidad de Sevilla. Departamento de álgebra |
Publication Date | 2007 |
Deposit Date | 2016-06-08 |
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Abstract | In this paper we study rank one discrete valuations of the field k((X1, . . . , Xn)) whose center in k[[X1, . . . , Xn]] is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations ... In this paper we study rank one discrete valuations of the field k((X1, . . . , Xn)) whose center in k[[X1, . . . , Xn]] is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use Zorn’s lemma or to know explicitly a section σ of the natural homomorphism Rv → ∆v between the ring and the residue field of the valuation v. However, as a consequence of this construction, in section 7, we prove that k((X1, . . . , Xn)) can be embedded into a field L((Y1, . . . , Yn)), where L is an algebraic extension of k and the “extended valuation” is as close as possible to the usual order function. |
Funding agencies | Ministerio de Educación y Ciencia (MEC). España European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER) |
Project ID. | MTM2004-07203-C02-01
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Citation | Herrera Govantes, F.J., Olalla Acosta, M.Á. y Vicente Córdoba, J.L. (2007). Rank one discrete valuations of power series fields. Communications in Algebra, 35 (8), 2533-2551. |
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