Abstract
We develop a modification of a notion of distance of an element in a valued field extension introduced by F.-V. Kuhlmann. We show that the new notion preserves the main properties of the distance and at the same time gives more complete information about a valued field extension. We study valued field extensions of prime degree to show the relation between the distances of the elements and the corresponding extensions of value groups and residue fields. In connection with questions related to defect extensions of valued function fields of positive characteristic, we present constructions of defect extensions of rational function fields K(x, y)|K generated by elements of various distances from K(x, y). In particular, we construct dependent Artin–Schreier defect extensions of K(x, y) of various distances.
Highlights
In this paper, we consider fields equipped with (Krull) valuations
If (L|K, v) is a valued field extension, every element z ∈ L induces a cut of the divisible hull v K of the value group of K in the following way
The cut in v K having as its lower cut set the smallest initial segment of v K containing {v(z − c) | c ∈ K } ∩ v K is denoted by dist K (z, K )
Summary
We consider fields equipped with (Krull) valuations. A valued field will be denoted by (K , v), its value group by v K , and its residue field by K v. The proof of Theorem 4.7 of the present paper shows in particular a possible construction of a valuation of a rational function field K (x, y)|K with residue field K and a given value group. 4 we use Theorem 4.7 to give a construction of Artin–Schreier defect extensions of the rational function field K (x, y) such that the distance dist (θ, K (x, y)) of the Artin–Schreier generator θ from K (x, y) is a given negative real number. This in particular gives constructions of Artin–Schreier defect extensions of rational function fields in two variables in the case of non- pdivisible value groups
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