Abstract
AbstractWe introduce the notion of approximation type for the partial, and in certain cases the total description of extensions of a given valuation from a field K to the rational function field K(x). To every extension, a unique approximation type of x over K is associated, while x may be the limit of many pseudo Cauchy sequences. Approximation types also provide information in cases where the extensions are not immediate, and we prove that they correspond bijectively to the extensions when K is algebraically closed or, more generally, lies dense in its algebraic closure with respect to the topology induced by the valuation.
Highlights
As we will show that under certain natural conditions, these extensions are uniquely determined by what we call approximation types, it is important to note from the start that we always identify equivalent valuations
In order to be able to compute the value of every element of K (x) with respect to v, it suffices to be able to compute the value of all polynomials in x, that is, we only have to deal with the polynomial ring K [x]
It is desirable to have a unique object that can readily be assigned to an element in a valued field extension and that contains all information that is contained in pseudo Cauchy sequences, and possibly more
Summary
In this paper we will work with (Krull) valuations on fields and their extensions to rational function fields. It is desirable to have a unique object that can readily be assigned to an element in a valued field extension and that contains all information that is contained in pseudo Cauchy sequences, and possibly more. We will define such objects, called approximation types, in Sect. Still it should be noted that like these sequences, the approximation types cannot describe all extensions when K is not algebraically closed. The classification of extensions we introduced above is clearly reflected in the approximation types, together with related information necessary to fully describe the valuations. Note that for simplicity we will often write “v” in place of “v0” for the valuation on K when it is not necessary to distinguish it from its extensions
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