A model-theoretic approach to the arithmetic of global fields

Thesis

This 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.

Authors: Dittmann, P

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Keywords: Field Arithmetic; Definability in Fields; Model Theory of Fields
• Subjects: Mathematics