  Let $K$ be an imaginary quadratic field with discriminant $d_K\leq-7$. We
deal with problems of constructing normal bases between abelian extensions of
$K$ by making use of singular values of Siegel functions. First, we show that a
criterion achieved from the Frobenius determinant relation enables us to find
normal bases of ring class fields of orders of bounded conductors depending on
$d_K$ over $K$. Next, denoting by $K_{(N)}$ the ray class field modulo $N$ of
$K$ for an integer $N\geq2$ we consider the field extension
$K_{(p^2m)}/K_{(pm)}$ for a prime $p\geq5$ and an integer $m\geq1$ relatively
prime to $p$ and then find normal bases of all intermediate fields over
$K_{(pm)}$ by utilizing Kawamoto's arguments. And, we further investigate
certain Galois module structure of the field extension
$K_{(p^{n}m)}/K_{(p^{\ell}m)}$ with $n\geq 2\ell$, which would be an extension
of Komatsu's work.
