For $C_2$ the cyclic group of order 2, I want to calculate $\tilde{K}_{0}(\mathbb{Z}[C_2])$. Now so far, I know by a theorem of Rim that $\tilde{K}_{0}(\mathbb{Z}[C_2])$ is isomorphic to the ideal class group of the cyclotomic field $\mathbb{Q}(\zeta_2)$, which has class number 1. So, we know $\tilde{K}_{0}(\mathbb{Z}[C_2])=0$. At least I am fairly sure this is right.
Now, is there any way to generalise this argument? Are there certain prime numbers for which we can always do this?
Rim's theorem says that, for a prime $p$, $\tilde {K_0} (\mathbf Z C_p) \cong \tilde {K_0} (\mathbf Z [\zeta_p]) \cong Cl (\mathbf Z [\zeta_p])$ , where the last term denotes the ideal class group of the $p$-th cyclotomic field, which in general is not trivial. The ideal class group has received much attention since Kummer's work on FLT. Kummer showed that if $p$ does not divide the class number of $\mathbf Q [\zeta_p]$, then FLT holds for the exponent $p$. But this is not the case in general, and it can even be said that the study of the class group is at the origin of the development of algebraic number theory. Much progress has been made, but many conjectures remain unsolved, e.g. Vandiver's conjecture, which predicts that $p$ does not divide the class number of the maximal real subfield of $\mathbf Q [\zeta_p]$ .
Rim's theorem says that $\tilde {K_0} (\mathbf Z C_p) \cong \tilde {K_0} (\mathbf Z [\zeta_p]) \cong Cl (\mathbf Z [\zeta_p])$ , where the last term denotes the ideal class group of the $p$-th cyclotomic field. The ideal class group has received much attention since Kummer's work on FLT. Kummer showed that if $\mathbf Z [\zeta_p]$ is principal, then FLT holds. But this is not the case in general, and it can even be said this is at the origin of the development of algebraic number theory.
Technical mistake: the first draft (before editing) has not been erased)