Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic
Zapiski Nauchnykh Seminarov POMI, Computational complexity theory. Part VIII, Tome 304 (2003), pp. 141-167

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Let $M$ be a field of positive characteristic $p>0$ such that $C$, the closure of a finite field in $M$, has an extension of degree $p$. Let $L$ be a field finitely generated over $C$ and such that $M$ and $L$ are linearly disjoint over $C$. Then Hilbert's Tenth problem is not decidable over $ML$.
@article{ZNSL_2003_304_a8,
     author = {A. Shlapentokh},
     title = {Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {141--167},
     publisher = {mathdoc},
     volume = {304},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_304_a8/}
}
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A. Shlapentokh. Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic. Zapiski Nauchnykh Seminarov POMI, Computational complexity theory. Part VIII, Tome 304 (2003), pp. 141-167. http://geodesic.mathdoc.fr/item/ZNSL_2003_304_a8/