Undecidability in some field theories

Thesis

This 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?

Authors: Tyrrell, B

• DOI: 10.5287/ora-aepoj7br8
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: hereditary undecidability; undecidability; PAC fields; henselian valued fields; PRC fields; function fields; Hilbert's tenth problem
• Subjects: Algebraic number theory; Model Theory; Undecidability