Computable fields and the bounded Turing reduction

For a computable field F , the splitting set SF of F is the set of polynomials in F [X] which factor over F , and the root set RF of F is the set of polynomials in F [X] which have a root in F . Results of Fröhlich 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, if we use a finer measure, 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 bounded Turing (bT) reduction. We construct a computable algebraic field for which RF bT SF . We also define a Rabin embedding g of a field into its algebraic closure, and for a computable algebraic field F , we compare the relative complexities of RF , SF , and g(F ) under m-reducibility and under bTreducibility.