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研究生： 高智強
Kao, Chih-Chiang
論文名稱： Iterated Galois Groups over Quadratic Number Field
Iterated Galois Groups over Quadratic Number Field
指導教授： 夏良忠
Hsia, Liang-Chung
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2020
畢業學年度： 108
語文別： 英文
論文頁數： 40
英文關鍵詞： iterated polynomial, arboreal Galois group, iterated wreath product, 2-independent
DOI URL： http://doi.org/10.6345/NTNU202000666
論文種類： 學術論文
相關次數： 點閱：152  下載：43 
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    • Consider the base field $K$ is a real quadratic number field and a polynomial $X^2+c$ where $c$ lies in the ring of integer $\mathcal{O}_K$. We will give some criteria on the iterated polynomial $f^n(X)$ of $X^2+c$ to determine whether the Galois group of $f^n(X)$ over $K$ is isomorphic to the wreath product of cyclic group of order $2$. Next, we will focus on the following three cases:
    \begin{enumerate}
    \item $K = \mathbb{Q}(\sqrt{2})$;
    \item $K = \mathbb{Q}(\sqrt{2p})$ where $p$ is a prime and $p\equiv 3
    mod 4$;
    \item $K = \mathbb{Q}(\sqrt{p})$ where $p$ is a prime and $p\equiv 1
    mod 4$.
    \end{enumerate}
    The class number of $\qq(\sqrt{2})$ is one, for the other two cases, we need to assume $h_K = 1$. We will give sufficient conditions on $c$ such that the Galois group of the iterated polynomial over $K$ is isomorphic to the iterated wreath product. In the last part, we will prove some $2$-independent property of an integer set over a quadratic number field.

    Contents Abstract i 1 Introduction 1 2 Preliminaries 3 3 Criteria for $\Omega_n\cong [C_2]^n 10 4 Iteration sequences associated to even integer polynomials 17 5 Some values of $c$ with $|b_i|\not\in K^2$ for all $i\geq 2$ 18 5.1 Case1: $K = \mathbb{Q}(\sqrt{2})$ 18 5.2 Case2: $K = \mathbb{Q}(\sqrt{2p})$ and $p\equiv 3 mod 4$ 21 5.3 Case3: $K = \mathbb{Q}(\sqrt{p})$ and $p\equiv 3 mod 4$ 23 6 $\Omega_n\cong [C_2]^n$ for $K = \mathbb{Q}(\sqrt{2})$ 27 7 $2$-independent property of integers over quadratic number field 31 References 39

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