Given $\mathbb{K}$ local non archimedean field, how can I find an example of two totally ramified extensions of $\mathbb{K}$ whose compositum is not totally ramified?
I know that every such extension is generated by a uniformizer, but I don't know how to start attacking the problem. Any hint?
Hint: Try extending $\Bbb{Q}_2$ by adjoining two distinct cube roots of two. If you adjoin both of them you also get something that gives an unramified extension.