Let $K\to L$ be an algebraic field extension and $\nu$ a valuation of $K$.
The purpose of this paper is to describe the totality of extensions
$\left\{\nu'\right\}$ of $\nu$ to $L$ using a refined version of MacLane's key
polynomials. In the basic case when $L$ is a finite separable extension and $rk
\nu=1$, we give an explicit description of the limit key polynomials (which can
be viewed as a generalization of the Artin--Schreier polynomials). We also give
a realistic upper bound on the order type of the set of key polynomials.
Namely, we show that if $char K=0$ then the set of key polynomials has order
type at most $\mathbb N$, while in the case $char K=p>0$ this order type is
bounded above by $([\log_pn]+1)\omega$, where $n=[L:K]$. Our results provide a
new point of view of the the well known formula
$\sum\limits_{j=1}^se_jf_jd_j=n$ and the notion of defect
Let (R, m, k) be a local noetherian domain with field of fractions K and Rν a valuation ring, dominating R (not necessarily birationally). Let ν|K:K* ↠Γ be the restriction of ν to K by definition, ν|K is centred at R. Let R^denote the m‐adic completion of R. In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace R by its m‐adic completion R^ and ν by a suitable extension ν^− to R^/P for a suitably chosen prime ideal P, such that P∩R=(0). The purpose of this paper is to give, assuming that R is excellent, a systematic description of all such extensions ν^− and to identify certain classes of extensions which are of particular interest for applications.
This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.
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.
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