3-rank of ambiguous class groups of cubic Kummer extensions

Let $$k=k_0(\root 3 \of {d})$$ k = k 0 ( d 3 ) be a cubic Kummer extension of $$k_0=\mathbb {Q}(\zeta _3)$$ k 0 = Q ( ζ 3 ) with $$d>1$$ d > 1 a cube-free integer and $$\zeta _3$$ ζ 3 a primitive third root of unity. Denote by $$C_{k,3}^{(\sigma )}$$ C k , 3 ( σ ) the 3-group of ambiguous classes of the extension $$k/k_0$$ k / k 0 with relative group $$G={\text {Gal}}(k/k_0)=\langle \sigma \rangle $$ G = Gal ( k / k 0 ) = ⟨ σ ⟩ . The aims of this paper are to characterize all extensions $$k/k_0$$ k / k 0 with cyclic 3-group of ambiguous classes $$C_{k,3}^{(\sigma )}$$ C k , 3 ( σ ) of order 3, to investigate the multiplicity m ( f ) of the conductors f of these abelian extensions $$k/k_0$$ k / k 0 , and to classify the fields k according to the cohomology of their unit groups $$E_{k}$$ 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 $$E_{k}$$ E k , and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results.