Capitulation in the Absolutely Abelian Extensions of some Number Fields II

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• 作者：Abdelmalek Azizi ; Abdelkader Zekhnini ; Mohammed Taous
• 关键词：Absolute and relative genus fields ; Fundamental systems of units ; 2 ; class group ; Capitulation ; Quadratic fields ; Biquadratic fields ; Multiquadratic CM ; fields
• 刊名：Acta Mathematica Vietnamica
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：42
• 期：1
• 页码：81-97
• 全文大小：
• 刊物主题：Mathematics, general;
• 出版者：Springer Singapore
• ISSN：2315-4144
• 卷排序：42

文摘

We study the capitulation of 2-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields \(\mathbb {k} =\mathbb {Q}(\sqrt {pq_{1}q_{2}}, i)\), where \(i=\sqrt {-1}\) and q₁≡class="EmphasisTypeItalic ">q₂≡−p≡−1 (mod 4) are different primes. For each of the three quadratic extensions \(\mathbb {K}/\mathbb {k}\) inside the absolute genus field 𝕜(∗) of 𝕜, we compute the capitulation kernel of \(\mathbb {K}/\mathbb {k}\). Then we deduce that each strongly ambiguous class of \(\mathbb {k}/\mathbb {Q}(i)\) capitulates already in 𝕜(∗).