Existence of a number field containing $\alpha$ where $p$ is unramified I want to better understand the structure of algebraic number fields and for this purpose I am thinking of various problems. One of them I cannot solve, is the following:
Let $p$ be some prime number and $\alpha \in \mathbb{C}$ integral over $\mathbb{Z}$. Is there always some algebraic number field $K$ such that $\alpha$ is in the ring of integers $O_K$ of $K$ and $p$ is unramified in $O_K$, i.e. if $pO_K = \prod_{i=1}^n P_i^{e_i}$ where $P_i$ are different prime ideals of $O_K$, then $e_i = 1$ for all $i$.

This is not always possible. We have:

Let $L/K/F$ be a tower of number fields and $p \subset F$ a prime. TFAE:
  
  
*
  
*$p$ is unramified in $L$;
  
*$p$ is unramified in $K$ and all primes of $K$ above $p$ are unramified in $L$.
  

This is a consequence of unique factorization of ideals.
In particular, in the situation of your question, if $p$ is unramified in some field containing $\alpha$ then $p$ is unramified in $\mathbb Q(\alpha)$. So to give a counterexample, it suffices to give an example of a number field $\mathbb Q(\alpha)$ and a prime that is ramified in it! Example:

 $\alpha = i$ and $p = 2$.

