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3-rank of ambiguous class groups of cubic Kummer extensions
Periodica Mathematica Hungarica ( IF 0.8 ) Pub Date : 2020-03-19 , DOI: 10.1007/s10998-020-00326-1
S. Aouissi , D. C. Mayer , M. C. Ismaili , M. Talbi , A. Azizi

Let $$k=k_0(\root 3 \of {d})$$ k = k 0 ( d 3 ) be a cubic Kummer extension of $$k_0=\mathbb {Q}(\zeta _3)$$ k 0 = Q ( ζ 3 ) with $$d>1$$ d > 1 a cube-free integer and $$\zeta _3$$ ζ 3 a primitive third root of unity. Denote by $$C_{k,3}^{(\sigma )}$$ C k , 3 ( σ ) the 3-group of ambiguous classes of the extension $$k/k_0$$ k / k 0 with relative group $$G={\text {Gal}}(k/k_0)=\langle \sigma \rangle $$ G = Gal ( k / k 0 ) = ⟨ σ ⟩ . The aims of this paper are to characterize all extensions $$k/k_0$$ k / k 0 with cyclic 3-group of ambiguous classes $$C_{k,3}^{(\sigma )}$$ C k , 3 ( σ ) of order 3, to investigate the multiplicity m ( f ) of the conductors f of these abelian extensions $$k/k_0$$ k / k 0 , and to classify the fields k according to the cohomology of their unit groups $$E_{k}$$ E k as Galois modules over G . The techniques employed for reaching these goals are relative 3-genus fields, Hilbert norm residue symbols, quadratic 3-ring class groups modulo f , the Herbrand quotient of $$E_{k}$$ E k , and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results.

中文翻译:

三次 Kummer 扩展的模糊类群的 3 秩

令 $$k=k_0(\root 3 \of {d})$$ k = k 0 ( d 3 ) 是 $$k_0=\mathbb {Q}(\zeta _3)$$ k 0 的三次库默扩展= Q ( ζ 3 ) 其中 $$d>1$$ d > 1 是一个无立方体的整数,$$\zeta _3$$ ζ 3 是一个原始的三次单位根。用$$C_{k,3}^{(\sigma )}$$ C k , 3 ( σ ) 表示扩展$$k/k_0$$ k / k 0 的含糊类的3-group与相对群$$G={\text {Gal}}(k/k_0)=\langle \sigma \rangle $$ G = Gal ( k / k 0 ) = ⟨ σ ⟩ . 本文的目的是用循环 3 组模糊类 $$C_{k,3}^{(\sigma )}$$ C k , 3 来表征所有扩展 $$k/k_0$$k / k 0 ( σ ) 3 阶,研究这些阿贝尔扩展 $$k/k_0$$ k / k 0 的导体 f 的多重性 m ( f ),并根据其单位群 $ 的上同调对场 k 进行分类$E_{k}$$ E k 作为 G 上的伽罗瓦模块。用于实现这些目标的技术是相对 3 属域、Hilbert 范数残差符号、二次 3 环类群模 f、$$E_{k}$$E k 的 Herbrand 商和中心正交幂等。所有的理论成果都以广泛的计算结果为基础。
更新日期:2020-03-19
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