Published 2023-07-01
“…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 $K$. …”
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