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:172 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.
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