This is an announcement of the following result. Let \(p\) be a fixed odd prime number, let \(K\) be a number field containing the \(p\)-th roots of unity and let \(C_ p\) be a cyclic group of order \(p\). In this paper the quotient is considered of the group of all realizations of \(C_ p\) as a Galois group over \(K\) and the subgroup of those realizations which have a relative normal integral basis. A description, in terms of power series attached to \(p\)-adic \(L\)- functions, is given of the Galois module structure of this quotient when \(K\) runs over all layers of the cyclotomic \(\mathbb{Z}_ p\)-extensions of a certain imaginary abelian field.