In algebraic number theory, at least in terms of a serious interesting result with a difficult proof, I'd say the Tchebotarev Density Theorem.
More elementary would be Sum(ef)=n and/or properties of decomposition and inertia groups. Also Dirichlet's Unit Theorem, depending on what order you work in.
In analytic number theory, I would say the Prime Number Theorem, at least if we're looking for the first difficult, interesting result.
