Bertrands 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 Bertrands 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 $x$.