Variations on a theme of Minkowski and Serre

If α is a root of unity in an integral domain O of characteristic zero, (α - 1 )^k ∈ nO, and no prime divisor of n is a unit in O, then α = 1 if n is a positive integer outside a finite set determined by k. We prove this result and generalizations of it, and give results when n is an element of the finite exceptional set. We give applications to endomorphisms of semi-abelian varieties, compatible systems of l-adic representations, and the cohomology of projective varieties.