Variations on a theme of Minkowski and Serre Original Research Article

If α is a root of unity in an integral domain image of characteristic zero, (α − 1)k ε nimage, and no prime divisor of n is a unit in image, 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.