  Let $K_{m}=\Bbb{Q}(\zeta_{m})$ where $\zeta_{m}$ is a primitive $m$th root of
unity. Let $p>2$ be prime and let $C_{p}$ denote the group of order $p.$ The
ring of algebraic integers of $K_{m}$ is $\Cal{O}_{m}=\Bbb{Z}[\zeta_{m}].$ Let
$\Lambda_{m,p}$ denote the order $\Cal{O}_{m}[C_{p}]$ in the algebra
$K_{m}[C_{p}].$ Consider the kernel group $D(\Lambda_{m,p})$ and the Swan
subgroup $T(\Lambda_{m,p}).$ If $(p,m)=1$ these two subgroups of the class
group coincide. Restricting to when there is a rational prime $p$ that is prime
in $\Cal{O}_{m}$ requires $m=4$ or $q^{n}$ where $q>2$ is prime. For each such
$m$, $3 \leq m \leq 100,$ we give such a prime, and show that one may compute
$T(\Lambda_{m,p})$ as a quotient of the group of units of a finite field. When
$h_{mp}^{+}=1$ we give exact values for $|T(\Lambda_{m,p})|$, and for other
cases we provide an upper bound. We explore the Galois module theoretic
implications of these results.
