I'm looking for an example of degree $n$ ramified but not totally ramified example over $\Bbb Q_p$.
I can find degree $n$ ramified extension, for example, $\Bbb Q_p({p^{1/n}})/\Bbb Q_p$. $p^{1/n}$'s minimal polynomial over $\Bbb Q_p$ is $x^n-p$ ($p$-eisenstein polynomial) and this is degree $n$, so the extension degree is exactly $n$. And norm of ${p^{1/n}}$ is $-p$, this is prime element of $\Bbb Q_p$, so the extension is totally ramified.
But this kind of construction only generates totally ramified cases.. But I'm looking for a degree $n$ and ramified but not totally ramified.
Could you give me an example of degree $n$ ramified but not totally ramified example over $\Bbb Q_p$?
If $n$ is prime, then any extension will be either unramified or totally ramified, because $n=ef$. Suppose that $n$ is not prime and $n=ml$ with $m,l>1$. Take a degree $m$ totally ramified extension, say $\Bbb Q_p(p^{1/m})$ and a degree $l$ unramified extension $\Bbb Q_p(\zeta_{w})$ where $w=p^l-1$. Then we can take the composite field $\Bbb Q_p(p^{1/m},\zeta_{w})$ this has degree $ml=n$. Indeed $x^m-p$ is is still Eisenstein over the field $\Bbb Q_p(\zeta_w)$ because $p$ is a prime element in the ring of integers (as $\Bbb Q_p(\zeta_w)/\Bbb Q_p$ is unramified). Because the ramification index and inertia degree are multiplicative, this extension has ramification index $m$ and inertia degree $l$, so it's ramified, but not totally ramified.