Decidability in extensions of Fp((t))

Thesis

In 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.

Authors: Rigler, B

• DOI: 10.5287/ora-good6pb61
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: extremal valued fields; model theory; logic; decidability; valuation theory
• Subjects: mathematics