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Preprint Number 1882

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1882. Anand Pillay and Erik Walsberg
Galois groups of large fields with simple theory

Submission date: 19 November 2020


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 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 PAC field L with the same absolute Galois group as K.

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Full text arXiv 2011.10018: pdf, ps.

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