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.

  • 1
    $\begingroup$ Ok, it works! Thanks for the tip! $\endgroup$ – Angelo Rendina Dec 24 '14 at 16:23
  • $\begingroup$ Glad to hear that. Well done, Angelo! $\endgroup$ – Jyrki Lahtonen Dec 24 '14 at 16:30

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