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

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).