Siham Aouissi, Daniel C. Mayer, Moulay Chrif Ismaili, Mohamed Talbi, Abdelmalek Azizi

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

Let $\mathrm{k}=\mathbb{Q}(\sqrt[3]{d},\zeta_3)$ be the Galois closure of a pure cubic field $\mathbb{Q}(\sqrt[3]{d})$, where $d>1$ is a cube-free positive integer and $\zeta_3$ is a primitive third root of unity. Denote by $C_{\mathrm{k},3}^{(\sigma)}$ the $3$-group of ambiguous ideal classes of the cubic Kummer extension $\mathrm{k}/\mathbb{Q}(\zeta_3)$ with relative group $G=\operatorname{Gal}(\mathrm{k}/\mathbb{Q}(\zeta_3))=\langle\sigma\rangle$. The aims of this paper are to determine all integers $d$ such that $\operatorname{rank}\,(C_{\mathrm{k},3}^{(\sigma)})=1$, to investigate the multiplicity $m(f)$ of the conductors $f$ corresponding to these radicands $d$, and to classify the fields $\mathrm{k}$ according to the cohomology of their unit groups $E_{\mathrm{k}}$ as Galois modules over $G$. The techniques employed for reaching these goals are relative $3$-genus fields, Hilbert norm residue symbols, and quadratic $3$-ring class groups modulo $f$. All theoretical achievements are underpinned by extensive computational results.
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