Galois groups of large fields with simple theory

Anand Pillay, Erik Walsberg

Published 2020-11-19Version 1

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 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, if $K$ is additionally perfect then $K$ has trivial Brauer group, and if $v$ 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. These results give evidence towards the conjecture that large simple fields are bounded $\mathrm{PAC}$. Combining our results with a theorem of Lubotzky and van den Dries we show that if $K$ is also perfect then there is a bounded $\mathrm{PAC}$ field $L$ with the same absolute Galois group as $K$.