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Author: 高智強
Kao, Chih-Chiang
Thesis Title: Iterated Galois Groups over Quadratic Number Field
Iterated Galois Groups over Quadratic Number Field
Advisor: 夏良忠
Hsia, Liang-Chung
Degree: 碩士
Master
Department: 數學系
Department of Mathematics
Thesis Publication Year: 2020
Academic Year: 108
Language: 英文
Number of pages: 40
Keywords (in English): iterated polynomial, arboreal Galois group, iterated wreath product, 2-independent
DOI URL: http://doi.org/10.6345/NTNU202000666
Thesis Type: Academic thesis/ dissertation
Reference times: Clicks: 206Downloads: 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

    [1]R. W. K. Odoni, Realising wreath products of cyclic groups as Galois groups. Mathe-matika, 35(1), 101–113 (1988).
    [2]M. Stoll, Galois groups overQof some iterated polynomials, Arch. Math. (Basel) 59,239-244 (1992).
    [3]K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, secondedition, GTM 84, Springer-Verlag, New York,
    [4]J. J. Rotman, An Introduction to the Theory of Groups, forth edition, GTM 148,Springer-Verlag, New York, 1995, ch.7.
    [5]J. Neukirch, Algebraic number theory, first edition, Springer-Verlag, Berlin Heidelberg,1999.
    [6]E. Horowitz and S. Sahni, Fundamentals of Computer Algorithms, first edition, Com-puter Science Press, 1978.
    [7]M. F. Atiyah and I. G. Macdonald, Introduction to Commutaive Algebra, Addison-Wesley, 1969.
    [8]R. W. K. Odoni, The Galois theory of composites and iterates of polynomials, Proc.London Math. Soc. 51(3), 385-414 (1985).
    [9]H. Hasse, Über mehrklassige, aber eingeschlechtige reell-quadratische Zahlkörper. Elem.Math., 20, 49–59 (1965).
    [10]R. Jones, The density of prime divisors in the arithmetic dynamics of quadratic poly-nomials. J. Lond. Math. Soc. 78(2), 523–544 (2008).

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