Galois groups of large simple fields

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 PAC field L with the same absolute Galois group as K . In the appendix we show that if K is large and NSOP ∞ 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 SOP ∞ (without assuming largeness).