Valuations in fields of power series. (English) Zbl 1055.12004

Let \(k\) be an algebraically closed field and \(R=k[[x_{1},\dots,x_{n}]]\) be the formal power series ring in the variables \(x_{i}\) and \(K\) be its quotient field. This paper deals with discrete valuations of \(K\) which are trivial on \(k\) and whose center at \(R\) is the maximal ideal generated by \(x_{1},\dots,x_{n}.\) The authors give explicitly a description of all discrete rank one valuations of \(k((x_{1},x_{2}))/k\) and \(k((x_{1},x_{2},x_{3}))/k.\) Constructive examples of rank two valuations are also given. The description is different from the one already known in the function field case [cf. S. Khanduja and U. Garg, Mathematika 37, 97–105 (1990; Zbl 0689.12018)].


12J10 Valued fields
13A18 Valuations and their generalizations for commutative rings


Zbl 0689.12018
Full Text: DOI arXiv EuDML


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