We use the Low Basis Theorem of Jockusch and Soare to show that all computable algebraic fields are d-computably categorical for a particular Turing degree d with d'=0'', but that not all such fields are 0'-computably categorical. We also prove related results about algebraic fields with splitting algorithms, and fields of finite transcendence degree over ℚ.
"d-computable categoricity for algebraic fields." J. Symbolic Logic 74 (4) 1325 - 1351, December 2009. https://doi.org/10.2178/jsl/1254748694