| Title:
|
Bicyclic commutator quotients with one non-elementary component (English) |
| Author:
|
Mayer, Daniel C. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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148 |
| Issue:
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2 |
| Year:
|
2023 |
| Pages:
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149-180 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $K$. (English) |
| Keyword:
|
Hilbert $3$-class field tower |
| Keyword:
|
maximal unramified pro-$3$ extension |
| Keyword:
|
unramified cyclic cubic extensions |
| Keyword:
|
Galois action |
| Keyword:
|
imaginary quadratic fields |
| Keyword:
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bicyclic $3$-class group |
| Keyword:
|
punctured capitulation types |
| Keyword:
|
statistics |
| Keyword:
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pro-$3$ groups |
| Keyword:
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finite $3$-groups |
| Keyword:
|
generator rank |
| Keyword:
|
relation rank |
| Keyword:
|
Schur $\sigma $-groups |
| Keyword:
|
low index normal subgroups |
| Keyword:
|
kernels of Artin transfers |
| Keyword:
|
abelian quotient invariants |
| Keyword:
|
$p$-group generation algorithm |
| Keyword:
|
descendant trees |
| Keyword:
|
antitony principle |
| MSC:
|
11R11 |
| MSC:
|
11R20 |
| MSC:
|
11R29 |
| MSC:
|
11R32 |
| MSC:
|
11R37 |
| MSC:
|
11Y40 |
| MSC:
|
20D15 |
| MSC:
|
20E18 |
| MSC:
|
20E22 |
| MSC:
|
20F05 |
| MSC:
|
20F12 |
| MSC:
|
20F14 |
| idZBL:
|
Zbl 07729570 |
| idMR:
|
MR4585574 |
| DOI:
|
10.21136/MB.2022.0127-21 |
| . |
| Date available:
|
2023-05-04T17:56:07Z |
| Last updated:
|
2023-09-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151682 |
| . |
| Reference:
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