Article
MSC:
11R11,
11R20,
11R29,
11R32,
11R37,
11Y40,
20D15,
20E18,
20E22,
20F05,
20F12,
20F14 | MR 4585574 | Zbl 07729570 | DOI: 10.21136/MB.2022.0127-21
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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
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
Summary:
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 $K$.
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