Bertrand's Postulate for Number Fields

Consider an algebraic number field, $K$, and its ring of integers, $\mathcal{O}_K$. There exists a smallest $B_K>1$ such that for any $x>1$ we can find a prime ideal, $\mathfrak{p}$, in $\mathcal{O}_K$ with norm $N(\mathfrak{p})$ in the interval $[x,B_Kx]$. This is a generalization of Bertrand's postulate to number fields, and in this paper we demonstrate that having a good asymptotic estimate for the number of ideals in $\mathcal{O}_K$ less than $x$ can produce an upper bound for $B_K$ in terms of the invariants of $K$. We compare the bounds obtained via this technique to what can be obtained from an effective prime ideal theorem.