  We study the problem of bounding the least prime that does not split
completely in a number field. This is a generalization of the classic problem
of bounding the least quadratic non-residue. Here, we present two distinct
approaches to this problem. The first is by studying the behavior of the
Dedekind zeta function of the number field near 1, and the second by relating
the problem to questions involving multiplicative functions. We derive the best
known bounds for this problem for all number fields with degree greater than 2.
We also derive the best known upper bound for the residue of the Dedekind zeta
function in the case where the degree is small compared to the discriminant.
