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.

[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).