Abstract:In this work, we study the integral algebraic K-theory groups of ring spectra whose homotopy groups are polynomial algebras in a single variable, including Morava K-theories and their two-periodic counterparts. Using a filtration on topological cyclic homology that we call the orbit filtration, we prove results on the cardinalities of these groups. When the coefficient field is the algebraic closure of Fp, we determine these cardinalities completely. For Morava K-theories, we combine these methods with the motivic filtration on topological cyclic homology to determine the cardinalities of the integral algebraic K-theory groups in every degree not congruent to 0 or 1 modulo 2p-2. This is joint work with Gabriel Angelini-Knoll.