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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.

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 generalize the main result of [K–K–S] concerning the convex embeddings of a chain Γ in a lexicographic power Δ Γ . For a fixed non-empty chain Δ, we derive necessary and sufficient conditions for the existence of non-empty solutions Γ to each of the lexicographic functorial equations (Δ Γ ) ≤0 � Γ , (Δ Γ ) � Γa nd (Δ Γ )

We characterize linearly ordered sets, abelian groups and fields that are symmetrically complete, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and fields are important because generalizations of Banach’s Fixed Point Theorem hold in them. We prove that symmetrically complete ordered abelian groups and fields are divisible Hahn products and real closed power series fields, respectively. This gives us a direct route to the construction of symmetrically complete ordered abelian groups and fields, modulo an analogous construction at the level of ordered sets; in particular, this gives an alternative approach to the construction of symmetrically complete fields in [12].

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 describe the valuation theoretic properties of the Hardy fields associated to models of $T(\exp)$ , where T is the theory of a polynomially bounded o-minimal expansion of the reals and $\exp$ is the real exponential function. We deduce that $T(\exp)$ has levels with parameters and is exponentially bounded. We establish a maximality property of $H(\mathbb{R}_{\rm an, powers})$ , the Hardy field of the expansion by the restricted analytic functions and power functions.

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.