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

TL;DR

This paper characterizes the higher even K-groups of rings of integers in number fields using class groups of extensions, extending prior work on K_2 and connecting to classical criteria for totally real fields.

Contribution

It provides a new algebraic K-theoretic formulation for higher even K-groups of rings of integers and generalizes existing results on K_2 to higher K-groups.

Findings

Describes higher even K-groups in terms of class groups of extensions.

Provides an algebraic K-theoretical formulation of Greenberg and Kida's criterion.

Extends previous results from K_2 to higher K-groups.

Abstract

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