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 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 $\mathrm{PAC}<!--FUNCTION\; APPLICATION-->$ field $L$ with the same absolute Galois group as $K$. In the appendix we show that if $K$ is large and ${\mathrm{NSOP}<!--FUNCTION\; APPLICATION-->}_{\infty }$ and $v$ is a nontrivial 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 ${\mathrm{SOP}<!--FUNCTION\; APPLICATION-->}_{\infty }$ (without assuming largeness).

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