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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 $K$ isadditionally perfect then $K$ has trivial Brauer group. These results giveevidence towards the conjecture that large simple fields are bounded PAC.Combining our results with a theorem of Lubotzky and van den Dries we show thatthere is a bounded $\mathrm{PAC}$ field $L$ with the same absolute Galois groupas $K$. 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 closedHenselization, so in particular the residue field of $(K,v)$ is algebraicallyclosed and the value group is divisible. The appendix also shows that formallyreal and formally $p$-adic fields are $\mathrm{SOP}_\infty$ (without assuminglargeness).

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