Construction of class fields over cyclotomic fields

Let l and p be odd primes. For a positive integer mu, let k(mu), be the ray class field of k = Q(e(2 pi i/l)) modulo 2p(mu). We present certain class fields K-mu, of k such that k(mu) subset of K mu subset of k(mu+1), and we provide a necessary and sufficient condition for K-mu = k(mu+1). We also construct, in the sense of Hilbert, primitive generators of the field K-mu over k(mu) by using Shimura's reciprocity law and special values of theta constants.