In Algebraic Number Theory we have a standard setting: let $A$ be a Dedekind domain, $K$ its fraction field, $L/K$ be a finite separable extension and $B$ be the integral closure of $A$ in $L$.
Since $B$ is also a Dedekind domain, one of the basic problems is: given a prime $\mathfrak{p}$ of $A$, find the factorization of $\mathfrak{p}B$ into prime ideals. A lot of the theory is based on simplifying this problem. For example, if $L/K$ is Galois, then Hilbert's ramification theory shows that this problem becomes a lot easier.
I think I understood the basic results of completion (for example, both the ramification index and the residual degree stay the same upon completion, and we always have that $B=A[\alpha]$ after completing) but I have yet to see an example of this being used in a concrete problem. I searched on the internet and on some books and found absolutely nothing.
I would appreciate seeing an example of the use of completion to solve a concrete problem (perhaps the basic problem described above) in number theory.