Bicyclic commutator quotients with one non-elementary component

Mayer, Daniel C.

Mathematica Bohemica, Tome 148 (2023) no. 2, pp. 149-180

Résumé

For any number field $K$ with non-elementary $3$-class group ${\rm Cl}_3(K)\simeq C_{3^e}\times C_3$, $e\ge 2$, the punctured capitulation type $\varkappa (K)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le 4$, 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 $G/G^\prime \simeq C_{3^e}\times C_3$, $2\le e\le 6$, 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 $K$.

Détail

DOI : 10.21136/MB.2022.0127-21

Classification : 11R11, 11R20, 11R29, 11R32, 11R37, 11Y40, 20D15, 20E18, 20E22, 20F05, 20F12, 20F14
Keywords: Hilbert $3$-class field tower; maximal unramified pro-$3$ extension; unramified cyclic cubic extensions; Galois action; imaginary quadratic fields; bicyclic $3$-class group; punctured capitulation types; statistics; pro-$3$ groups; finite $3$-groups; generator rank; relation rank; Schur $\sigma $-groups; low index normal subgroups; kernels of Artin transfers; abelian quotient invariants; $p$-group generation algorithm; descendant trees; antitony principle