Kernel Groups and nontrivial Galois module structure of imaginary quadratic fields

Abstract

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.