On the integrality of the Witten–Reshetikhin–Turaev 3-manifold invariants

We prove that the SU(2) Witten–Reshetikhin–Turaev invariant of any 3-manifold with any colored link inside at any root of unity is an algebraic integer. As a byproduct, we get a new proof of the integrality of the SO(3) Witten–Reshetikhin–Turaev invariant for any 3-manifold with any colored link inside at any root of unity of odd order.