Galois groups of large fields with simple theory (with an appendix by Philip Dittmann)

TL;DR

This paper investigates large fields with simple theories, proving they are bounded and have certain algebraic properties, and explores their Galois groups and valuation theory, providing evidence for conjectures about their structure.

Contribution

It establishes that large simple fields are bounded and have trivial Brauer groups under certain conditions, and links their Galois groups to bounded PAC fields, advancing understanding of their model-theoretic and algebraic properties.

Findings

Large simple fields are bounded with finitely many separable extensions.

Any genus 0 curve over such fields has a rational point.

If perfect, these fields have trivial Brauer group.

Abstract

Suppose that $K$ is an infinite field which is large (in the sense of Pop) and whose first order theory is simple. We show that $K$ is {\em bounded}, namely has only finitely many separable extensions of any given finite degree. We also show that any genus $0$ curve over $K$ has a $K$-point and if $K$ is additionally perfect then $K$ has trivial Brauer group. These results give evidence towards the conjecture that large simple fields are bounded PAC. Combining our results with a theorem of Lubotzky and van den Dries we show that there is a bounded $\mathrm{PAC}$ field $L$ with the same absolute Galois group as $K$. In the appendix we show that if $K$ is large and $\mathrm{NSOP}_\infty$ and $v$ is a non-trivial valuation on $K$ then $(K,v)$ has separably closed Henselization, so in particular the residue field of $(K,v)$ is algebraically closed and the value group is divisible. The appendix also shows that formally real and formally $p$-adic fields are $\mathrm{SOP}_\infty$ (without assuming largeness).