Construction of class fields over imaginary biquadratic fields

Let $K$ be an imaginary biquadratic field, $K_1$, $K_2$ be its imaginary quadratic subfields and $K_3$ be its real quadratic subfield. For integers $N>0$, $\mu\geq 0$ and an odd prime $p$ with $\gcd(N,p)=1$, let $K_{(Np^\mu)}$ and $(K_i)_{(Np^\mu)}$ for $i=1,2,3$ be the ray class fields of $K$ and $K_i$, respectively, modulo $Np^\mu$. We first present certain class fields $\widetilde{K_{N,p,\mu}^{1,2}}$ of $K$, in the sense of Hilbert, which are generated by ray class invariants of $(K_i)_{(Np^{\mu+1})}$ for $i=1,2$ over $K_{(Np^\mu)}$ and show that $K_{(Np^{\mu+1})}=\widetilde{K_{N,p,\mu}^{1,2}}$ for almost all $\mu$. And we shall further construct a primitive generator of the composite field $K_{(Np^\mu)}(K_3)_{(Np^{\mu+1})}$ over $K_{(Np^\mu)}$ by means of norms of the above ray class invariants, which is a real algebraic integer. Using this value we also generate a primitive generator of $(K_3)_{(p)}$ over the Hilbert class field of the real quadratic field $K_3$, and further find its normal basis.