  We construct a new infinite family of pairs of imaginary cyclic fields of
degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime
number $p$. For the proof, we use the fundamental unit of $\mathbb
Q(\sqrt{p})$, certain units which are roots of a parametric quartic polynomial,
the Kummer theory, the Gauss sums and the Jacobi sums, linear recurrence
sequences, a consequence of the Weil conjecture and a result of Lenstra which
is a generalization of Artin conjecture on primitive roots. Our result is based
on the famous Scholz' results on pairs of quadratic fields $\mathbb
Q(\sqrt{D})$ and $\mathbb Q (\sqrt{-3D})$.
