Hilbert's Tenth Problem over Function Fields of Positive Characteristic Not Containing the Algebraic Closure of a Finite Field
Abstract
We prove that the existential theory of any function field K of characteristic p> 0 is undecidable in the language of rings provided that the constant field does not contain the algebraic closure of a finite field. We also extend the undecidability proof for function fields of higher transcendence degree to characteristic 2 and show that the first-order theory of any function field of positive characteristic is undecidable in the language of rings without parameters.
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