Exercise 6, § 2

Define an increasing sequence of algebraic number fields $K_0 = \mathbf{Q}, K_1, \ldots, K_n, \ldots$ such that, if $A_n$ denotes the integral closure of $\mathbf{Z}$ in $K_n$, $p$ a prime number $\neq 2$, $K_n$ is a quadratic extension of $K_{n-1}$ and there exists in $A_n$ a prime ideal $p_n$ lying above $(p)$ such that there are in $A_{n+1}$ two distinct prime ideals $p_{n+1}, p'_n$, lying above $p_n$ (observe that, in a finite field of characteristic $\neq 2$, there are always elements which are not squares). Let $K$ be the union of the $K_n$ and $A'$ the integral closure of $\mathbf{Z}$ in $K$; show that there exists an infinity of (maximal) ideals of $A'$ lying above the maximal ideal $(p)$ of $\mathbf{Z}$.