I just read the proof of this theorem that $\mathbb{Q}_p$ has finitely many totally ramified extensions of any degree $n$. The proof uses Krasner's lemma and the compactness of a space which corresponds to Eisenstein polynomials of degree $n$. One then picks a finite subcover which represents every possible such extension.
This proof technique is not very useful if one actually wants to count the number of totally ramified extensions of a particular degree $n$. Does anyone know of any actual methods for computing this?