Hilbert’s Tenth Problem for Algebraic Function Fields of Characteristic 2

Let K be an algebraic function field of characteristic 2 with constant field CK . Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree 2. Assume that there are elements u, x of K with u transcendental over CK and x algebraic over C(u) and such that K = CK(u, x). Then Hilbert’s Tenth Problem over K is undecidable. Together with Shlapentokh’s result for odd characteristic this implies that Hilbert’s Tenth Problem for any such field K of finite characteristic is undecidable. In particular, Hilbert’s Tenth Problem for any algebraic function field with finite constant field is undecidable.