# Example of degree $n$ ramified, but not totally ramified extension

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.
• @dandelion when I said "p-adic field", I meant any finite extension of $\Bbb Q_p$ (so any local field of char 0). It's true that the ring of integers of $\Bbb Q_p(\zeta_w)$ is a DVR, too. Commented Sep 22, 2021 at 16:39
• @dandelion I think that should be in every book covering local fields, e.g. Neukirch's Algebraic Number Theory. The basic idea is that for a finite extension $L/\Bbb Q_p$, there's a unique extension of the p-adic valuation to $L$ and the ring of integers of $L$ is the valuation ring associated to that valuation. But the valuation remains discrete, so the valuation ring is a DVR, hence a PID, hence a UFD. Commented Sep 22, 2021 at 18:08
• @dandelion discrete means that the value group is isomorphic to $\Bbb Z$, if $L$ is a finite extension of $\Bbb Q_p$ with ramification index $e$, then the extended valuation will have value group $\frac{1}{e}\Bbb Z$, but that's still isomorphic to $\Bbb Z$. Commented Sep 23, 2021 at 5:21