56 research outputs found
On the decidability of finite extensions of decidable fields
No full textThis paper is primarily concerned with the following question which first appeared in Koenigsmannâs On a Question of Abraham Robinsonâs: Is a finite field extension of a decidable field always decidable?
This offers a âtwistâ to a question that was originally posed by Abraham Robinson in 1973 which had asked whether every finitely generated extension of an undecidable field remains undecidable. The above-mentioned work of Koenigsmann from 2016 (and independently, a result of Cherlin, van den Dries, and Macintyre much earlier in the 1980s) showed that there are indeed undecidable fields which admit decidable finite extensions. This paper aims to show that one could, similarly, find examples of decidable fields which admit undecidable finite extensions, thereby answering the above-stated question negatively.
This result is achieved by identifying a sufficient condition which a decidable field must satisfy in order for it to have an undecidable finite extension. In an earlier iteration of this work, we had pointed out what we had believed to be one such condition. Unfortunately, this turned out not to be the case, which we illustrate using an explicit example. Through this demonstration, we were able to accentuate the weakness of the formerly mentioned criterion, which we strengthen in this thesis. We provide justification that the strengthened criterion is indeed sufficient â any decidable field satisfying this strengthened criterion would form a counterexample to the above-mentioned question.
We study one such class of decidable fields, known as the wonderful extensions of the rational numbers, first introduced by Ershov in the early 2000s, whose (sufficiently saturated) elementary extensions satisfy this strengthened criterion. This provides us with a concrete counterexample which shows that there are indeed decidable fields which admit undecidable finite extensions. We also point out various attempts at finding other counterexamples to the above-mentioned question, the difficulties faced in those instances, and some further questions in the flavour of the above-mentioned question that appear to be interesting in their own rights.</p
Definable henselian valuations and absolute Galois groups
No full textThis thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study. We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Since the existence of a parameter-free definable henselian valuation on a field ensures that henselianity is elementary in , we also study henselianity as an -property. Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields
Definable henselian valuations and absolute Galois groups
No full textThis thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study.
We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Since the existence of a parameter-free definable henselian valuation on a field ensures that henselianity is elementary in , we also study henselianity as an -property.
Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.This thesis is not currently available in ORA
On Galois correspondences in formal logic
No full textThis thesis examines two approaches to Galois correspondences in formal logic. A standard result of classical first-order model theory is the observation that models of L-theories with a weak form of elimination of imaginaries hold a correspondence between their substructures and automorphism groups defined on them. This work applies the resultant framework to explore the practical consequences of a model-theoretic Galois theory with respect to certain first-order L-theories. The framework is also used to motivate an examination of its underlying model-theoretic foundations. The model-theoretic Galois theory of pure fields and valued fields is compared to the algebraic Galois theory of pure and valued fields to point out differences that may hold between them. The framework of this logical Galois correspondence is also applied to the theory of pseudoexponentiation to obtain a sketch of the Galois theory of exponential fields, where the fixed substructure of the complex pseudoexponential field B is an exponential field with the field Qrab as its algebraic subfield. This work obtains a partial exponential analogue to the Kronecker-Weber theorem by describing the pure field-theoretic abelian extensions of Qrab, expanding upon work in the twelfth of Hilbert’s problems. This result is then used to determine some of the model-theoretic abelian extensions of the fixed substructure of B. This work also incorporates the principles required of this model-theoretic framework in order to develop a model theory over substructural logics which is capable of expressing this Galois correspondence. A formal semantics is developed for quantified predicate substructural logics based on algebraic models for their propositional or nonquantified fragments. This semantics is then used to develop substructural forms of standard results in classical first-order model theory. This work then uses this substructural model theory to demonstrate the Galois correspondence that substructural first-order theories can carry in certain situations
Model theory and algebra of positive characteristic Hahn fields
No full textThis thesis mainly explores model theoretic and algebraic properties of positive characteristic Hahn fields. We show that any positive characteristic tame Hahn field F((t^Γ )) containing t is decidable in Lt, the language of valued fields with a constant symbol for t, if F and Γ are decidable. In particular, we obtain decidability of Fp((t^{1/p^∞})) and Fp((t^Q)) in Lt. This uses a new AKE-principle for equal characteristic tame fields in Lt, building on work by Kuhlmann, together with Kedlaya’s work on finite automata and algebraic extensions of function fields. In the process, we obtain an AKE-principle for tame fields in mixed characteristic and extend a theorem by Rayner on the relative algebraic closure of function fields inside Hahn fields. Furthermore, for any Hahn field containing Fp((t)), we use an approximation method described originally by Lampert to obtain a bound on the order type of elements that are algebraic over Fp((t))
Undecidability in some field theories
No full textThis thesis is a study of undecidability in some field theories. Specifically, we are interested in geometrically oriented problems and have focused our attention in two directions along these lines. The first direction bases on determining the decidability of certain sets of first-order sentences over positive characteristic function fields. We will draw parallel to the problem of algorithmically determining in some cases the existence of points on varieties in positive characteristic function fields; equivalently the existence of certain maps between varieties over other positive characteristic fields.
The second direction bases on determining the decidability of first-order consequences of nonempty finite collections of L_r-sentences, true in fields with plenty of geometric structure. This is connected to the former direction by the fact that a decidable field has a recursive axiomatisation – what if we study a (nonempty) finite subset of the axiomatisation? Undecidability results.
Motivated by classification-theoretic conjectures, we will examine ‘wilder’ classes of fields in turn and generalise a result of Ziegler to NIP henselian nontrivially valued fields (and beyond). We move to PAC & PRC fields and prove they are finitely undecidable, resolving two open questions of Shlapentokh & Videla, and describe the difficulties that arise in adapting the proof to PpC fields. We pose the question: is every infinite field finitely undecidable
Definability in Henselian fields
No full textWe investigate definability in henselian fields. Specifically, we are interested in those sets and substructures that are existentially definable or definable with 'few' parameters. Our general approach is to use various versions of henselianity to understand the 'local structure' of these definable sets. The fields in which we are most interested are those of positive characteristic, for example the local fields Fq((t)), but many of our methods and results also apply to p-adic and real closed fields. In positive characteristic we have to deal with inseparable field extensions and we develop the method of Λ-closure to `translate' inseparable field extensions into separable ones. In the first part of the thesis we focus on existentially definable sets, which are projections of algebraic sets. Our main tool is the Implicit Function Theorem (for polynomials) which is equivalent to t-henselianity, by work of Prestel and Ziegler. This enables us to prove that existentially definable sets are `large' in various senses. Using the Implicit Function Theorem, we also obtain a nonuniform local elimination of the existential quantifier. The non-uniformity and local character of this result at present forms an obstacle to full quantifier-elimination. From these technical statements we can deduce characterisations of, for example, existentially definable subfields and existentially definable transcendentals. We prove that a dense, regular extension of t-henselian fields is existentially closed which, in particular, implies the old result of Ershov that Fp(t)h ≤Ǝ Fp((t)). Using the existential closedness of large fields in henselian fields, we are able to apply many of these results to large fields. This answers questions for imperfect large fields that were answered in the perfect case by Fehm. In the second part of the thesis, we work with power series fields F((t)) and subsets which are F- definable (and not contained in F). We use a `hensel-like' lemma to characterise F-orbits of (singleton) elements of F((t)). It turns out that all such orbits are Ǝ-t-definable. Consequently, we may apply our earlier results about existentially definable subsets to F-definable subsets. We can use this to characterise F-definable subfields of F((t)). As a further corollary, we obtain an Ǝ-0̸-definition of Fp[[t]] in Fp((t))
A model-theoretic approach to the arithmetic of global fields
No full textThis thesis assembles some new results in the field arithmetic of various classes of fields, including global fields, models of the common first-order theory of algebraic extensions of global fields, and fields of finite transcendence degree over their prime field. Most of the results stem from a very simple technique for first-order definitions in fields, based on the satisfaction of a first-order sentence in a family of finite extensions of the ground field. In the first two chapters, we develop this technique to associate existentially definable sets to central simple algebras and
Pfister forms, respectively.
We then use these tools to obtain results on global fields, their algebraic extensions, fields elementarily equivalent to ultraproducts thereof, and finitely generated fields. We study valuations on such fields, and notably obtain a large class of examples of fields without Self-Embedded Residue, a natural notion that arises in the study of definable valuations.
Subsequently, we focus on the study of a single global field, where we obtain new definability results. Most importantly, we show that non-solubility of a polynomial equation in one variable over the global field is expressible as an existential condition on the coefficients. This also yields consequences in algebraic geometry over the given field.
After a category-theoretic interlude in the model theory of absolute Galois groups, the final chapter investigates p-valuations on fields. We introduce a notion of generalised Stufe in this context, and prove an interpretability result for spaces of p-valuations in situations of interest.</p
Decidability in extensions of Fp((t))
No full textIn this thesis we primarily consider the first-order theory of the local field F_p((t)) and the question of whether it is decidable. To this end we expand the language of valued fields together with a constant symbol for t by adding predicates R_f representing the existence of a root of the multivariable polynomial f \in F_p(t)[X_1, ..., X_n], hoping in this way to control the behaviour of purely wild extensions of valued fields (which has historically caused problems in this area), and seek to prove a quantifier elimination result for a proposed axiomatisation of F_p((t)) (originally, we believe, due to F-V. Kuhlmann). This proceeds along classical lines, à la Shoenfield (used in Macintyre’s celebrated proof of quantifier elimination for the p-adic numbers Q_p). We obtain a conditional result, sketching a proof that if a particular technical condition is satisfied by an arbitrary \aleph_1-saturated model of our theory, then we will indeed obtain quantifier elimination and thereby (via the existence of an algebraically prime
model) decidability of the first-order theory of F_p((t)). However, this technical condition is still largely mysterious. We sketch an (unsuccessful) attempt at an unconditional proof and highlight where difficulties arise.
In Chapter 1 we introduce the area in general, and our question and approach more specifically. Chapter 2 contains necessary preliminaries for the rest of the thesis, on valuation theory, model theory, Galois theory, and a smidgen of algebraic geometry; we have intended to keep these preliminaries as brief as possible. Chapter 3 starts with an explanation of our approach and contains sections (among others) introducing and analysing our candidate axiomatisation, proceeding to a conditional Shoenfield-style quantifier elimination proof (Theorem 3.4.2) and presenting (in \S 3.5) a sketch of where difficulties arise when trying an unconditional proof. Chapter 4 introduces several notions which we feel may be helpful in future study of this area, as well as discussing ultraproducts of generalised Laurent series fields in relation to extremality. Chapter 5 includes a self-contained examination of the structure and first-order theory of some distinguished fields extending F_p((t)), this time
with p-divisible value group. We conclude with some brief remarks in Chapter 6
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