We use a known example of an algebraically maximal discretely valued field of positive characteristic p which admits purely inseparable extensions of degree p^2 with defect p to construct algebraically maximal valued fields of characteristic p as well as of characteristic 0 and of rank 2 which admit separable extensions of degree p^2 with defect p.

For an algebraic function field F over a large field K, we show: 1) if F|K has a rational place, then there is a finite purely inseparable extension K'|K such that K' is existentially closed in F.K'; 2) F|K has a rational place admitting local uniformization if and only if K is existentially closed in F.

We investigate the existence, uniqueness and maximality of solutions T for equations S1+T=S2 and inequalities S1+T subset of S2, where S1 and S2 are final segments of ordered abelian groups. Since cuts are determined by their upper cut sets, which are final segments, this gives information about the corresponding equalities and inequalities for cuts. We apply our results to investigate the existence, uniqueness and maximality of solutions J for equations I1J=I2 and inequalities I1J subset of I2, where I1 and I2 are ideals of valuation rings. This enables us to compute the annihilators of quotients of the form I1/I2.

For important cases of algebraic extensions of valued fields, we develop presentations of the associated K\"ahler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated Dedekind differentials. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs.\ independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, K\"ahler differentials and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their rank or value groups.

We give a survey on recent developments in the model theory of valued fields since the introduction of the notion of “tame valued field”, and of the modifications and generalizations of this notion.

We introduce and study the notion of ramification ideals in higher ramification theory. After general results on their computation for finite extensions, we discuss their connection with the possibly nontrivial defect of the extensions. We compute them for Artin-Schreier extensions and Kummer extensions of prime degree equal to the residue characteristic, which may or may not have nontrivial defect. We present an example that shows that nontrivial defect in an extension of degree p^2, p a prime, may not imply the existence of a nonprincipal ramification ideal.

We show how suitable extensions (L|K,v) of prime degree of valued fields give rise to definable coarsenings of the valuation rings of L and K. In the case of Artin-Schreier and Kummer extensions with wild ramification, we can also define the ramification ideal. We demonstrate the use of the coarsenings on L, their maximal ideals, and the ramification ideals for the classification of defects and for the presentation of the Kähler differentials of the extension of the valuation rings of (L|K,v), and their annihilators. Finally, we give a construction that realizes predescribed convex subgroups of suitable value groups as those that are associated with Galois extensions of degree p with independent defect, which in turn give rise to definable coarsenings.

We study spherical completeness of ball spaces and its stability under expansions. We give some criteria for ball spaces that guarantee that spherical completeness is preserved when the ball space is closed under unions of chains. This applies in particular to the spaces of closed ultrametric balls in ultrametric spaces with linearly ordered value sets, or more generally, with countable narrow value sets. We show that in general, chain union closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness. Further, we introduce and study the notions of chain union stability and of chain union rank, which measure how often the process of closing a ball space under all unions of chains has to be iterated until a ball space is obtained that is closed under unions of chains.

We study spherical completeness of ball spaces and its stability under expansions. We give some criteria for ball spaces that guarantee that spherical completeness is preserved when the ball space is closed under unions of chains. This applies in particular to the spaces of closed ultrametric balls in ultrametric spaces with linearly ordered value sets, or more generally, with countable narrow value sets. We show that in general, chain union closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness. Further, we introduce and study the notions of chain union stability and of chain union rank, which measure how often the process of closing a ball space under all unions of chains has to be iterated until a ball space is obtained that is closed under unions of chains.

Given an algebraic function field F|K$F|K$ and a place P on K, we prove that the places that are composite with extensions of P to finite extensions of K lie dense in the space of all places of F, in a strong sense. We apply the result to the case of K=R$K=R$ any real closed field and the fixed place on R being its natural (finest) real place. This leads to a new description of the real holomorphy ring of F, which can be seen as an analog to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space M(F)$M(F)$ of all R$\mathbb {R}$-places of F (places with residue field contained in R$\mathbb {R}$), its subspace of all R$\mathbb {R}$-places of F that are composite with the natural R$\mathbb {R}$-place of R, and the topological space of all R-rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper, the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space M(F)$M(F)$ has only finitely many topological components.

We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed elements in its algebraic closure, with the extensions generated by them forming an increasing chain. In the case of algebraic extensions, we generalize the results to countably generated infinite tame extensions over henselian but not necessarily tame fields. In the case of transcendental extensions, we demonstrate the central role that is played by the implicit constant fields, which reveals the tight connection with the algebraic case.

We study in detail the valuation theory of deeply ramified fields and introduce and investigate several other related classes of valued fields. Further, a classification of defect extensions of prime degree of valued fields that was earlier given only for the equicharacteristic case is generalized to the case of mixed characteristic by a unified definition that works simultaneously for both cases. It is shown that deeply ramified fields and the other valued fields we introduce only admit one of the two types of defect extensions, namely the ones that appear to be more harmless in open problems such as local uniformization and the model theory of valued fields in positive characteristic. We use our knowledge about such defect extensions to give a new, valuation theoretic proof of the fact that algebraic extensions of deeply ramified fields are again deeply ramified. We also prove finite descent, and under certain conditions even infinite descent, for deeply ramified fields. These results are also proved for two other related classes of valued fields. The classes of valued fields under consideration can be seen as generalizations of the class of tame valued fields. Our paper supports the hope that it will be possible to generalize to deeply ramified fields several important results that have been proven for tame fields and were at the core of partial solutions of the two open problems mentioned above.

Assume that (L,v) is a finite Galois extension of a valued field (K,v). We give an explicit construction of the valuation ring 𝒪_L of L as an 𝒪_K-algebra, and an explicit description of the module of relative Kähler differentials Ω_𝒪_L|𝒪_K when L|K is a Kummer extension of prime degree or an Artin-Schreier extension, in terms of invariants of the valuation and field extension. The case when this extension has nontrivial defect was solved in a recent paper by the authors with Anna Rzepka. The present paper deals with the complementary (defectless) case. The results are known classically for (rank 1) discrete valuations, but our systematic approach to non-discrete valuations (even of rank 1) is new. We also show that the annihilator of Ω_𝒪_L|𝒪_K can only be equal to the maximal ideal ℳ_L of 𝒪_L if the extension is defectless and ℳ_L is principal. Using our results from the prime degree case, we characterize when Ω_𝒪_L|𝒪_K=0 holds for an arbitrary finite Galois extension of valued fields. As an application of these results, we give a simple proof of a theorem of Gabber and Ramero, which characterizes when a valued field is deeply ramified. We further give a simple characterization of deeply ramified fields with residue fields of characteristic p>0 in terms of the Kähler differentials of Galois extensions of degree p.

We prove that a valued field of positive characteristic $p$ that has only finitely many distinct Artin-Schreier extensions (which is a property of infinite NTP$_2$ fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has $p$-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin-Schreier defect extensions.

We classify cuts in (totally) ordered abelian groups Γ and compute the coinitiality and cofinality of all cuts in case Γ is divisible, in terms of data intrinsically associated to the invariance group of the cut. We relate cuts with small extensions of Γ in a natural way, which leads to an explicit construction of a totally ordered real vector space containing realizations of all cuts. This construction is applied to the problem of classifying all extensions of the valuation from a given valued field K to the rational function field K(x).

Using the ramification theory of tame and Kaplansky fields, we show that maximal Kaplansky fields contain maximal immediate extensions of each of their subfields. Likewise, algebraically maximal Kaplansky fields contain maximal immediate algebraic extensions of each of their subfields. This study is inspired by problems that appear in henselian valued fields of rank higher than 1 when a Hensel root of a polynomial is approximated by the elements generated by a (transfinite) Newton algorithm.

We 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.

Spherically complete ball spaces provide a framework for the proof of generic fixed point theorems. For the purpose of their application it is important to have methods for the construction of new spherically complete ball spaces from given ones. Given various ball spaces on the same underlying set, we discuss the construction of new ball spaces through set theoretic operations on the balls. A definition of continuity for functions on ball spaces leads to the notion of quotient spaces. Further, we show the existence of products and coproducts and use this to derive a topological category associated with ball spaces.

With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions of functions being contractive in these spaces and structures. As a sample of possible applications we discuss metric spaces, ultrametric spaces, ordered groups and fields, topological spaces, partially ordered sets, and lattices. We describe several notions of completeness in these spaces and structures and determine their respective strengths. In order to illustrate some consequences of the levels of strength, we give examples of generic fixed point theorems which then can be specialized to theorems in various applications which work with contracting functions and some completeness property of the underlying space. Ball spaces are nonempty sets of nonempty subsets of a given set. They are called spherically complete if every chain of balls has a nonempty intersection. This is all that is needed for the encoding of completeness notions. We discuss operations on the sets of balls to determine when they lead to larger sets of balls; if so, then the properties of the so obtained new ball spaces are determined. The operations can lead to increased level of strength, or to ball spaces of newly constructed structures, such as products. Further, the general framework makes it possible to transfer concepts and approaches from one application to the other; as examples we discuss theorems analogous to the Knaster–Tarski Fixed Point Theorem for lattices and theorems analogous to the Tychonoff Theorem for topological spaces.

We consider four approaches to the analysis of cuts in ordered abelian groups and ordered fields, their interconnection, and various applications. The notions we discuss are: ball cuts, invariance group, invariance valuation ring, and cut cofinality.