On the 2-class field tower of $\mathbb Q (\sqrt{2p_1p_2},i)$ and the Galois group of its second Hilbert 2-class field

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• 作者：Abdelmalek Azizi (1)
  Abdelkader Zekhnini (1)
  Mohammed Taous (2)
  
• 关键词：Class groups ; Class fields ; Class numbers ; Class field towers ; Hilbert class ; 11R27 ; 11R29 ; 11R37
• 刊名：Collectanea Mathematica
• 出版年：2014
• 出版时间：January 2014
• 年：2014
• 卷：65
• 期：1
• 页码：131-141
• 全文大小：327 KB
• 作者单位：Abdelmalek Azizi (1)
  Abdelkader Zekhnini (1)
  Mohammed Taous (2)
  
  1. D茅partement de Math茅matiques, Facult茅 des Sciences, Universit茅 Mohammed 1, Oujda, Morocco
  2. D茅partement de Math茅matiques, Facult茅 des Sciences et Techniques, Universit茅 Moulay Ismail, Errachidia, Morocco
  
• ISSN：2038-4815

文摘

Let $p_1$ and $p_2$ be primes such that $p_1\equiv p_2\equiv 5 \pmod 8$ , $i=\sqrt{-1}$ , $d=2p_1p_2$ , $\mathbb K =\mathbb Q (\sqrt{d},i)$ , $\mathbb K _2^{(1)}$ be the Hilbert 2-class field of $\mathbb K $ , $\mathbb K _2^{(2)}$ be the Hilbert 2-class field of $\mathbb K _2^{(1)}$ , $G$ be the Galois group of $\mathbb K _2^{(2)}/\mathbb K $ and $\mathbb K ^{(*)}=\mathbb Q (\sqrt{p_1},\sqrt{p_2},\sqrt{2}, i)$ be the genus field of $\mathbb K $ . The 2-part $\mathbf C _{\mathbb{K },2}$ of the class group of $\mathbb K $ is of type $(2, 2, 2)$ . Our goal is to study the 2-class field tower of $\mathbb K $ and to calculate the order of $G$ .