A Topological Quantum Field Theory (TQFT) is a functorial extension of invariants of 3-manifolds to manifolds with boundaries. They are thus highly structured and imply, for example, nontrivial representations of the mapping class groups. A large family of such TQFT\'s is given by the Witten-Reshetikhin-Turaev TQFT\'s. Assuming a mild modification of the TQFT axioms it is possible to define them over the cyclotomic integers (rather than just the complex numbers). The rich ideal structure of this ring combined with the modified functoriality yields a new and quite subtle tool to investigate various properties of the mapping class groups, specific 3-manifolds, and some of their classical invariants. In the talk I will give several examples of such applications.