Iterated Galois Groups over Quadratic Number Field

Abstract

none

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.