It is well-known what the $p^n$-cyclotomic extensions (i.e., adjoining $p^n$-th roots of unity) of $\mathbb{Q}_p$ are (see Serre, Local fields for instance).

However, assume now that $K/\mathbb{Q}_p$ is an arbitrary finite extension. What can now be said about the $p^n$-cyclotomic extensions of $K$?

It is clear that this is a harder problem and I haven't been able to find any literature on this but I'm sure that it's out there. At least some cases (for instance, the case $K/\mathbb{Q}_p$ unramified should be similar to the "classical case" I think).

**Edit**: I should have specified what I mean by "What can be said...?". What I'm interested in in particular is the ramification groups and the jumps in the ramification filtration.

/Daniel