Kernel Groups and Nontrivial Galois Module Structure of Imaginary Quadratic Fields

Let K be an algebraic number field with ring of integers OK , p > 2, be a rational prime and G the cyclic group of order p. Let Λ denote the order OK [G]. Let Cl(Λ) denote the locally free class group of Λ and D(Λ) the kernel group, the subgroup of Cl(Λ) consisting of classes that become trivial upon extension of scalars to the maximal order. If p is unramified in K, then D(Λ) = T (Λ), where T (Λ) is the Swan subgroup of Cl(Λ). This yields upper and lower bonds for D(Λ). Let R(Λ) denote the subgroup of Cl(Λ) consisting of those classes realizable as rings of integers, OL, where L/K is a tame Galois extension with Galois group Gal(L/K) ∼= G. We show under the hypotheses above that T (Λ)(p−1)/2 ⊆ R(Λ) ∩ D(Λ) ⊆ T (Λ), which yields conditions for when T (Λ) = R(Λ) ∩ D(Λ) and bounds on R(Λ) ∩ D(Λ). We carry out the computation for K = Q( √−d), d > 0, d = 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.