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

Abstract

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 $k/k_0$ with relative group $G=\mathrm{Gal}(k/k_0)=⟨\sigma ⟩$. The aims of this paper are to characterize all extensions $k/k_0$ 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 $k/k_0$, 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.

24 pages, 7 tables, 3 figures

Cited by in corpus (3)

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