Structure of relative genus fields of cubic Kummer extensions. (English) Zbl 1537.11132
Consider the number field \(N={\mathbb Q}(\sqrt[3]{D}, \zeta_3)\), a cubic Kummer extension of the cyclotomic field \(K={\mathbb Q}(\zeta_3)\), where \(D\) is a cube free positive integer and \(\zeta_3\) a primitive cubic root of unity.
The main result of the article establishes the exhaustive list of the 13 forms of the integer \(D\) –in terms of its prime decomposition– for which the Galois group over \(N\) of the relative 3-genus field of the extension \(N/K\) is isomorphic to \({\mathbb Z}/3{\mathbb Z} \times {\mathbb Z}/3{\mathbb Z}\).
The relative 3-genus field of the extension \(N/K\) is the maximal abelian extension of \(K\) contained in the maximal abelian unramified 3-extension of \(N\), and studying its Galois group over \(N\) is a first step into the determination of the structure of \(Cl_3(N)\), the 3-class group of the number field \(N\).
The authors extend in the present article the results of their previous paper [S. Aouissi et al., Period. Math. Hung. 81, No. 2, 250–274 (2020; Zbl 1474.11186)] based on Gerth’s methods, see [F. Gerth III, J. Reine Angew. Math. 278/279, 52–62 (1975; Zbl 0334.12011); J. Number Theory 8, 84–98 (1976; Zbl 0329.12006)]. Moreover, they also study and classify the different extensions \(N/K\) in terms of their conductor.
The main result of the article establishes the exhaustive list of the 13 forms of the integer \(D\) –in terms of its prime decomposition– for which the Galois group over \(N\) of the relative 3-genus field of the extension \(N/K\) is isomorphic to \({\mathbb Z}/3{\mathbb Z} \times {\mathbb Z}/3{\mathbb Z}\).
The relative 3-genus field of the extension \(N/K\) is the maximal abelian extension of \(K\) contained in the maximal abelian unramified 3-extension of \(N\), and studying its Galois group over \(N\) is a first step into the determination of the structure of \(Cl_3(N)\), the 3-class group of the number field \(N\).
The authors extend in the present article the results of their previous paper [S. Aouissi et al., Period. Math. Hung. 81, No. 2, 250–274 (2020; Zbl 1474.11186)] based on Gerth’s methods, see [F. Gerth III, J. Reine Angew. Math. 278/279, 52–62 (1975; Zbl 0334.12011); J. Number Theory 8, 84–98 (1976; Zbl 0329.12006)]. Moreover, they also study and classify the different extensions \(N/K\) in terms of their conductor.
Reviewer: Isabelle Dubois (Metz)
MSC:
| 11R11 | Quadratic extensions |
| 11R16 | Cubic and quartic extensions |
| 11R20 | Other abelian and metabelian extensions |
| 11R27 | Units and factorization |
| 11R29 | Class numbers, class groups, discriminants |
| 11R37 | Class field theory |
Keywords:
pure cubic field; cubic Kummer extension; relative genus field; group of ambiguous ideal classes; 3-rank; primitive ambiguous principal ideals; multiplicity of conductorsReferences:
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