For a finite-dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite-dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite-dimensional Hopf kernel and with the image of the universal enveloping algebra. In this article, we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases, the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.