  Let $K$ be an algebraic number field with ring of integers $\Cal{O}_{K}$,
$p>2$ be a rational prime and $G$ be the cyclic group of order $p $. Let
$\Lambda$ denote the order $\Cal{O}_{K}[G].$ Let $Cl(\Lambda)$ denote the
locally free class group of $\Lambda$ and $D(\Lambda)$ the kernel group, the
subgroup of $Cl(\Lambda)$ consisting of classes that become trivial upon
extension of scalars to the maximal order. If $p$ is unramified in $K$, then
$D(\Lambda) = T(\Lambda)$, where $T(\Lambda)$ is the Swan subgroup of
$Cl(\Lambda).$ This yields upper and lower bounds for $D(\Lambda)$. Let
$R(\Lambda)$ denote the subgroup of $Cl(\Lambda)$ consisting of those classes
realizable as rings of integers, $\Cal{O}_{L},$ where $L/K$ is a tame Galois
extension with Galois group $Gal(L/K) \cong G.$ We show under the hypotheses
above that $T(\Lambda)^{(p-1)/2} \subseteq R(\Lambda) \cap D(\Lambda) \subseteq
T(\Lambda)$, which yields conditions for when $T(\Lambda)=R(\Lambda) \cap
D(\Lambda)$ and bounds on $R(\Lambda) \cap D(\Lambda)$. We carry out the
computation for $K=\Bbb{Q}(\sqrt{-d}), d>0, d \neq 1$ or $3.$ In this way we
exhibit primes $p$ for which these fields have tame Galois field extensions of
degree $p$ with nontrivial Galois module structure.
