Iterated Galois Groups over Quadratic Number Field
dc.contributor | 夏良忠 | zh_TW |
dc.contributor | Hsia, Liang-Chung | en_US |
dc.contributor.author | 高智強 | zh_TW |
dc.contributor.author | Kao, Chih-Chiang | en_US |
dc.date.accessioned | 2020-12-14T09:00:11Z | |
dc.date.available | 2020-07-10 | |
dc.date.available | 2020-12-14T09:00:11Z | |
dc.date.issued | 2020 | |
dc.description.abstract | none | zh_TW |
dc.description.abstract | 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. | en_US |
dc.description.sponsorship | 數學系 | zh_TW |
dc.identifier | G060740017S | |
dc.identifier.uri | http://etds.lib.ntnu.edu.tw/cgi-bin/gs32/gsweb.cgi?o=dstdcdr&s=id=%22G060740017S%22.& | |
dc.identifier.uri | http://rportal.lib.ntnu.edu.tw:80/handle/20.500.12235/111255 | |
dc.language | 英文 | |
dc.subject | none | zh_TW |
dc.subject | iterated polynomial | en_US |
dc.subject | arboreal Galois group | en_US |
dc.subject | iterated wreath product | en_US |
dc.subject | 2-independent | en_US |
dc.title | Iterated Galois Groups over Quadratic Number Field | zh_TW |
dc.title | Iterated Galois Groups over Quadratic Number Field | en_US |
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