## On the structure of even $K$-groups of rings of algebraic integers

### Volume 211 / 2023

Acta Arithmetica 211 (2023), 345-361
MSC: Primary 11R70; Secondary 11R42, 19F27.
DOI: 10.4064/aa221029-25-7
Published online: 19 October 2023

#### Abstract

We describe the higher even $K$-groups of the ring of integers of a number field in terms of the class groups of an appropriate extension of the number field in question. This is a natural extension of the previous work of Browkin, Keune and Kolster, who considered the case of $K_2$. We then revisit Kummer’s criterion for totally real fields as generalized by Greenberg and Kida. In particular, we give an algebraic-$K$-theoretical formulation of the criterion which we prove using the algebraic-$K$-theoretical results developed here.