While Goodwillie's result holds for arbitrary rings, a quick reduction shows that it is in essence only a statement about formula_11. For rings not containing Q, cyclic homology must be replaced by topological cyclic homology in order to keep a close connection to K-theory. (If Q is contained in "A", then cyclic homology and topological cyclic homology of "A" agree.) This is in line with the fact that (classical) Hochschild homology is less well-behaved than topological Hochschild homology for rings not containing Q. proved a far-reaching generalization of Goodwillie's result, stating that for a commutative ring "A" so that the Henselian lemma holds with respect to the ideal "I", the relative K-theory is isomorphic to relative topological cyclic homology (without tensoring both with Q). Their result also encompasses a theorem of , asserting that in this situation the relative K-theory spectrum modulo an integer "n" which is invertible in "A" vanishes. used Gabber's result and Suslin rigidity to reprove Quillen's computation of the K-theory of finite fields.