  Let K be an algebraic function field of characteristic 2 with constant field
C_K. 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 C_K and x algebraic over C(u) and such that K=C_K(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.
