3-class field towers with 2 or 3 stages
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| Formato: | Preprint |
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2026
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| _version_ | 1866917454696939520 |
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| author | von Berghof, Helga Boyer Mayer, Daniel C. |
| author_facet | von Berghof, Helga Boyer Mayer, Daniel C. |
| contents | For quadratic fields \(k=\mathbb{Q}(\sqrt{d})\) with discriminant \(d\), \(3\)-class group \(\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2\), and four \textit{simple} \(3\)-principalization types \(\varkappa(k)\in\lbrace (1122),(3122),(1231),(2231)\rbrace\), we establish necessary and sufficient conditions for the Galois group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) of the unramified Hilbert \(3\)-class field tower of \(k\) to coincide with the Galois group \(M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)\) of the maximal metabelian unramified \(3\)-extension of \(k\). In the case of non-coincidence, we study the path between \(M\) and \(S\) in the descendant tree of the elementary bicyclic \(3\)-group \((\mathbb{Z}/3\mathbb{Z})^2\). For two \textit{complex} \(3\)-principalization types \(\varkappa(k)\in\lbrace (2122),(4231)\rbrace\), we show that infinitely many non-metabelian possible Galois groups \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) with presumably unbounded derived length \(\mathrm{dl}(S)\) share a common metabelianization \(M=S/S^{\prime\prime}\), whence only partial criteria can be stated. Minimal discriminants \(d>0\) with assigned simple \(3\)-principalization type \(\varkappa(k)\) and fixed length \(\ell_3(k)\in\lbrace 2,3\rbrace\) of the \(3\)-class field tower are determined experimentally for nilpotency class \(\mathrm{cl}(M)\in\lbrace 5,7,9,11\rbrace\) under assumption of the generalized Riemann hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01590 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 3-class field towers with 2 or 3 stages von Berghof, Helga Boyer Mayer, Daniel C. Number Theory 11R37, 11R29, 11R11, 11R16, 11R20, 20D15, 20F14 For quadratic fields \(k=\mathbb{Q}(\sqrt{d})\) with discriminant \(d\), \(3\)-class group \(\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2\), and four \textit{simple} \(3\)-principalization types \(\varkappa(k)\in\lbrace (1122),(3122),(1231),(2231)\rbrace\), we establish necessary and sufficient conditions for the Galois group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) of the unramified Hilbert \(3\)-class field tower of \(k\) to coincide with the Galois group \(M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)\) of the maximal metabelian unramified \(3\)-extension of \(k\). In the case of non-coincidence, we study the path between \(M\) and \(S\) in the descendant tree of the elementary bicyclic \(3\)-group \((\mathbb{Z}/3\mathbb{Z})^2\). For two \textit{complex} \(3\)-principalization types \(\varkappa(k)\in\lbrace (2122),(4231)\rbrace\), we show that infinitely many non-metabelian possible Galois groups \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) with presumably unbounded derived length \(\mathrm{dl}(S)\) share a common metabelianization \(M=S/S^{\prime\prime}\), whence only partial criteria can be stated. Minimal discriminants \(d>0\) with assigned simple \(3\)-principalization type \(\varkappa(k)\) and fixed length \(\ell_3(k)\in\lbrace 2,3\rbrace\) of the \(3\)-class field tower are determined experimentally for nilpotency class \(\mathrm{cl}(M)\in\lbrace 5,7,9,11\rbrace\) under assumption of the generalized Riemann hypothesis. |
| title | 3-class field towers with 2 or 3 stages |
| topic | Number Theory 11R37, 11R29, 11R11, 11R16, 11R20, 20D15, 20F14 |
| url | https://arxiv.org/abs/2605.01590 |