$3$-rank of ambiguous class groups of cubic Kummer extensions

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}({\zeta }_3)$ with $d>1$ a cube-free integer and ${\zeta }_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma )}$ the $3$-group of ambiguous classes of the extension $k\mathrm{/}{k}_0$ with relative group $G=\mathrm{Gal}(k\mathrm{/}{k}_0)=⟨\sigma ⟩$. The aims of this paper are to characterize all extensions $k\mathrm{/}{k}_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(\sigma )}$ of order $3$, to investigate the multiplicity $m(f)$ of the conductors $f$ of these abelian extensions $k\mathrm{/}{k}_0$, and to classify the fields $k$ according to the cohomology of their unit groups $E_k$ as Galois modules over $G$. The techniques employed for reaching these goals are relative $3$-genus fields, Hilbert norm residue symbols, quadratic $3$-ring class groups modulo $f$, the Herbrand quotient of $E_k$, and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results.