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arXiv · 2605.28299

Classification Theory and the Construction of PAC Fields

Nicholas Ramsey
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Abstract

We analyze a construction of Cherlin, van den Dries, and Macintyre to code graphs in PAC fields. We show that, in many cases, model-theoretic properties of the graph are preserved in the passage from the graph to the field. As a corollary, we show that the SOP$_{n}$ hierarchy is strict in the class of fields. The main ingredient is a detailed treatment of the model theory of inverse systems of certain profinite groups that code graphs and can be realized as the absolute Galois groups of PAC fields.

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Nicholas Ramsey. 2026-05-27. Classification Theory and the Construction of PAC Fields. https://arxiv.org/abs/2605.28299

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