  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 produce bounds on
$B_K$ in terms of the invariants of $K$ from an effective prime ideal theorem
due to Lagarias and Odlyzko. We also show that a bound on $B_K$ can be obtained
from an asymptotic estimate for the number of ideals in $\mathcal{O}_K$ less
than $x$.
