On even $K$-groups of rings of integers of real abelian fields

Abstract

We present approaches for calculating the precise orders of the algebraic $K$-groups $K_{4n-2}({\mathcal{O}}_K)$ for a totally real abelian $K$. Along the way, we also establish a formula connecting the order of $K_{4n-2}({\mathcal{O}}_E)$ of a totally real $p$-elementary field $E$ to its intermediate cyclic $p$-degree fields. Additionally, we provide a compiled list of values for these $K$-groups.