Milnor (page 29, see below) gives an explicit proof that the zeroth $K$-theory of the group ring $\mathbb{Z}[C_p]$, where $C_p$ is the cyclic group of order $p$ with $p$ a prime agrees with $K_0$ of $\mathbb{Z}[\zeta_p]$ where $\zeta_p$ is a $p$-th root of unity. The reduced $K$-theory of this group can in turn be identified with the ideal class group. Kummer has computed these back in the 19th century for small $p$ and the first prime where this group is non-trivial is $p = 23$, where the ideal class group is isomorphic to $\mathbb{Z}/3$. This is all well and good, but at the end of the day, I want to use these elements to construct explicit counterexamples in $K$-theory, which means I need to be able to compute things. Elements in $K_0$ should be representable by idempotent matrices. My questions are the following:
What is a concrete idempotent matrix over $\mathbb{Z}[C_{23}]$ representing a generator of the $\mathbb{Z}/3$-summand in $K_0$?
How can I use the computation of ideal class groups of cyclotomic rings of integers in general to describe the corresponding idempotent matrices in $K$-theory?
Milnor, John W., Introduction to algebraic K-theory, Annals of Mathematics Studies. No. 72. Princeton, N. J.: Princeton University Press and University of Tokyo Press. XIII, 184 p. $ 6.25 (1971). ZBL0237.18005.
