Is It Harder to Factor a Polynomial or to Find a Root?, Part II
Rebecca Steiner, Vanderbilt University
Location: Stevenson 1312
For a computable algebraic field F, the splitting set S_F of F is the set of polynomials with coefficients in F which factor over F, and the root set R_F of F is the set of polynomials with coefficients in F which have a root in F. In the first part of this talk, on October 2, 2012, we showed that under the bounded Turing (bT) reducibility, determining whether a polynomial has a root in a field F is more difficult than determining whether it factors over F, i.e. S_F is always bT-reducible to R_F, but there are fields F where R_F is not bT-reducible to S_F. In the second part, we will 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 R_F, S_F, and g(F) under m-reducibility and under bT-reducibility.