Coclass of the second 3-class group
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918186774953984 |
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| author | Aouissi, Siham Mayer, Daniel C. |
| author_facet | Aouissi, Siham Mayer, Daniel C. |
| contents | By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is determined unambiguously by the second largest order ord(Cl_3(E_2))=3^{cc(M)+1} among the four 3-class groups of the unramified cyclic cubic extensions E_i (i=1,..,4) of K. Minimal discriminants of quadratic and cubic fields K with assigned coclass cc(M) are computed from extensive databases of 3-class numbers ord(Cl_3(E_i)) as an application. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_17510 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coclass of the second 3-class group Aouissi, Siham Mayer, Daniel C. Number Theory 11R29, 11R37, 20D15 By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is determined unambiguously by the second largest order ord(Cl_3(E_2))=3^{cc(M)+1} among the four 3-class groups of the unramified cyclic cubic extensions E_i (i=1,..,4) of K. Minimal discriminants of quadratic and cubic fields K with assigned coclass cc(M) are computed from extensive databases of 3-class numbers ord(Cl_3(E_i)) as an application. |
| title | Coclass of the second 3-class group |
| topic | Number Theory 11R29, 11R37, 20D15 |
| url | https://arxiv.org/abs/2508.17510 |