Computable Fields and Weak Truth-Table Reducibility

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 [6], 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 .