Given a positive integer n and a finite extension $K$ of $\mathbb{Q}$, can one always find an irreducible polynomial in $K[x]$ of degree n? What if $n$ is prime?

The natural approach is to take a prime ideal $\mathfrak{p}$ in $\mathcal{O}_K$, choose an element $\alpha \in \mathfrak{p} - \mathfrak{p}^2$, and consider the polynomial $x^n - \alpha$. It is irreducible over $\mathcal{O}_K$ by Eisenstein's Criterion. If $\mathcal{O}_K$ is a Schreier domain, then Gauss's Lemma applies and $x^n - \alpha$ is irreducible over $K$. The problem is that $\mathcal{O}_K$ need not be a Schreier domain (and Schreier domains are exactly the domains where Gauss's Lemma works).