First-order definitions in function fields over anti-Mordellic fields

Details First-order definitions in function fields over anti-Mordellic fields First-order definitions in function fields over anti-Mordellic fields

2006-02-24

arxiv.org/abs/math/0602541v1

Mathematics

Number Theory

12 pages

Scientific paper

A field k is called anti-Mordellic if every smooth curve over k with a k-point has infinitely many k-points. We prove that for a function field over an anti-Mordellic field, the subfield of constants is defined by a certain universal first order formula. Under additional hypotheses regarding 2-cohomological dimension we prove that algebraic dependence of an n-tuple of elements in such a function field can be described by a first order formula, for each n. We also give a result that lets one distinguish various classes of fields using first order sentences.

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