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

A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by $p$