  We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an
arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a
rational surgery formula for the unified invariant dominating WRT SO(3)
invariants of rational homology 3-spheres at roots of unity of order co-prime
with the torsion. As an application, we compute the unified invariant for
Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this
invariant separates integral homology Seifert fibered spaces and can be used to
detect the unknot.
