Group theoretic approach to cyclic cubic fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909059962109952 |
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| author | Aouissi, Siham Mayer, Daniel C. |
| author_facet | Aouissi, Siham Mayer, Daniel C. |
| contents | Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary bicyclic, the automorphism group M = Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k is determined by conditions for cubic residue symbols between p,q,r and for ambiguous principal ideals in subfields of the common absolute 3-genus field k* of k1,k2,k3,k4. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G = Gal(F(3,infinity,k)/k) of the maximal unramified pro-3-extension of k. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07349 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Group theoretic approach to cyclic cubic fields Aouissi, Siham Mayer, Daniel C. Number Theory 11R16, 11R20, 11R27, 11R29, 11R37, 20D15 Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary bicyclic, the automorphism group M = Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k is determined by conditions for cubic residue symbols between p,q,r and for ambiguous principal ideals in subfields of the common absolute 3-genus field k* of k1,k2,k3,k4. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G = Gal(F(3,infinity,k)/k) of the maximal unramified pro-3-extension of k. |
| title | Group theoretic approach to cyclic cubic fields |
| topic | Number Theory 11R16, 11R20, 11R27, 11R29, 11R37, 20D15 |
| url | https://arxiv.org/abs/2310.07349 |