On the $p$-divisibility of even $K$-groups of the ring of integers of a cyclotomic field

TL;DR

This paper investigates conditions under which odd primes divide the orders of even K-groups of rings of integers in cyclotomic fields, providing both sufficient and necessary criteria for divisibility in specific cases.

Contribution

It offers new criteria for p-divisibility of K-groups of cyclotomic integer rings, including necessary and sufficient conditions in certain p-extensions.

Findings

Provides sufficient conditions for p dividing K-groups of cyclotomic integer rings.

Establishes necessary and sufficient conditions for p-divisibility in specific p-extensions.

Enhances understanding of the structure of K-groups in cyclotomic fields.

Abstract

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[\zeta_m+\zeta_m^{-1}])$ and $K_{2k}(\mathbb{Z}[\zeta_m])$, where $\zeta_m$ is a primitive $m$th root of unity. When $F$ is a $p$-extension contained in $\mathbb{Q}(\zeta_l)$ for some prime $l$, we also establish a necessary and sufficient condition for the order of $K_{2(p-2)}(\mathcal{O}_F)$ to be divisible by $p$.