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We prove in arbitrary characteristic that an immediate valued algebraic function field $F$ of transcendence degree 1 over a tame field $K$ is contained in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates ramification in such valued function fields. We give generalizations of this result, relaxing the assumption on $K$. Our theorems have important applications to local uniformization and to the model theory of valued fields in positive and mixed characteristic., new improved version

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

A basic version of Abhyankar's Lemma states that for two finite extensions $L$ and $F$ of a local field $K$, if $L|K$ is tamely ramified and if the ramification index of $L|K$ divides the ramification index of $F|K$, then the compositum $L.F$ is an unramified extension of $F$. In this paper, we generalize the result to valued fields with value groups of rational rank 1, and show that the latter condition is necessary. Replacing the condition on the ramification indices by the condition that the value group of $L$ be contained in that of $F$, we generalize the result further in order to give a necessary and sufficient condition for the elimination of tame ramification of an arbitrary extension $F|K$ by a suitable algebraic extension of the base field $K$. In addition, we derive more precise ramification theoretical statements and give several examples.

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields., 30 pages. arXiv admin note: substantial text overlap with arXiv:1304.0194

The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect extensions through the tool of distances, which measure how well an element in an immediate extension can be approximated by elements from the base field. We show that in several situations the number of essentially distinct distances in fixed extensions, or even just over a fixed base field, is finite, and we compute upper bounds. We apply this to the special case of valued functions fields over perfect base fields. In particular, this provides important information used in forthcoming research on the ramification theory of two-dimensional valued function fields.

A henselian valued field K is called a tame field if its algebraic closure K ~ ${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K ~ | K ${\tilde{K}|K}$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero., Comment: 12 pages

We study necessary and sufficient conditions for a valued field K with value group G and residue field k (with char K = char k ) to admit a truncation closed embedding in the field of generalized power series k ( ( G , f ) ) (with factor set f). We show that this is equivalent to the existence of a family (tower of complements) of k-subspaces of K which are complements of the (possibly fractional) ideals of the valuation ring, and satisfying certain natural conditions. If K is a Henselian field of characteristic 0 or, more generally, an algebraically maximal Kaplansky field, we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding. We also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions.

We study spaces $M(R(y))$ of $\R$-places of rational function fields $R(y)$ in one variable. For extensions $F|R$ of formally real fields, with $R$ real closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in $M(F(y))$ and prove uniqueness results. Further, we study embeddings of products of spaces of the form $M(F(y))$ in spaces of $\R$-places of rational function fields in several variables. Our results uncover rather unexpected obstacles to a positive solution of the open question whether the torus can be realized as a space of $\R$-places.

We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then wild ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author's proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.

We prove a general version of the "Stability Theorem": if $K$ is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions, and if $F|K$ is a finitely generated (transcendental) extension of valued fields for which equality holds in the Abhyankar inequality, then the defect is also trivial for all finite extensions of $F$. This theorem is applied to eliminate ramification in such valued function fields. It has applications to local uniformization and to the model theory of valued fields in positive characteristic., Comment: 31 pages

We consider the Zariski space of all places of an algebraic function field $F|K$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding problems over large fields}. We also study the question whether a field $K$ is existentially closed in an extension field $L$ if $L$ admits a $K$-rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space., Comment: 27 pages

For valued fields $K$ of rank higher than 1, we describe how elements in the henselization $K^h$ of $K$ can be approximated from within $K$; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of $K$. We apply the result to show that if an element $z$ algebraic over $K$ can be approximated from within $K$ in the same way as an element in $K^h$, then $K(z)$ is not linearly disjoint from $K^h$ over $K$., Comment: 13 pages

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed or algebraically closed field F with subfield K, we give a sufficient condition for a valued subfield of the field of generalized power series F((G)) to admit a K-valuation basis. We show that the field of rational functions F(G) and the field F(G)not, vert, similar of power series in F((G)) algebraic over F(G) satisfy this condition. It follows that for archimedean F and divisible G the real closed field F(G)not, vert, similar admits a restricted exponential function.

Using valuation rings and valued fields as examples, we discuss in which ways the notions of “topological IFS attractor” and “fractal space” can be generalized to cover more general settings.