Mathematica Bohemica, Vol. 148, No. 2, pp. 149-180, 2023
Bicyclic commutator quotients with one non-elementary component
Daniel C. Mayer
Received August 25, 2021. Published online May 3, 2022.
Abstract: For any number field $K$ with non-elementary 3-class group ${\rm Cl}_3(K)\simeq C_{3^e}\times C_3$, $e\ge2$, the punctured capitulation type $\varkappa(K)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le4$, is an orbit under the action of $S_3\times S_3$. By means of Artin's reciprocity law, the arithmetical invariant $\varkappa(K)$ is translated to the punctured transfer kernel type $\varkappa(G_2)$ of the automorphism group $G_2={\rm Gal}({\rm F}_3^2(K)/K)$ of the second Hilbert 3-class field of $K$. A classification of finite 3-groups $G$ with low order and bicyclic commutator quotient $G/G^\prime\simeq C_{3^e}\times C_3$, $2\le e\le6$, according to the algebraic invariant $\varkappa(G)$, admits conclusions concerning the length of the Hilbert 3-class field tower ${\rm F}_3^\infty(K)$ of imaginary quadratic number fields $K$.
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