Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Adapting Rabin's theorem for differential fields
CiE'11 Proceedings of the 7th conference on Models of computation in context: computability in Europe
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For a computable field F, the splitting set SF of F is the set of polynomials with coefficients in F which factor over F, and the root set RF of F is the set of polynomials with coefficients in F which have a root in F. Results of Frohlich and Shepherdson in [3] imply that for a computable field F, the splitting set SF and the root set RF are Turing-equivalent. Much more recently, in [5], Miller showed that for algebraic fields, the root set actually has slightly higher complexity: for algebraic fields F, it is always the case that SF ≤1 RF, but there are algebraic fields F where we have RF ≤1 SF. Here we compare the splitting set and the root set of a computable algebraic field under a different reduction: the weak truth-table reduction. We construct a computable algebraic field for which RF ≤wtt SF.