Subfields of ample fields I. Rational maps and definability

2008/11/18 by Fehm, Arno · 1 citation
#03C60 #12E30 #12F99 #14G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0811.2895

Abstract

Pop proved that a smooth curve C over an ample field K that has a K-rational point has |K| many K-rational points. We strengthen this result by showing that there are |K| many K-rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfect ample field has no existentially definable proper infinite subfields.

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