TITLE: About cyclotomic extensions of $p$-adic fields
QUESTION [15 upvotes]: I've been working on the problem of finding the maximal abelian extension of $\mathbb{Q}_5$ that is killed by $5$. In other words, find the abelian extension of $\mathbb{Q}_5$ with Galois group isomorphic to $\mathbb{Q}_5^\times/\mathbb{Q}_5^{\times 5}\simeq C_5\times C_5$. Now
$$(\mathbb{Q}_5^\times :\mathbb{Q}_5^{\times 5})=25$$
so essentially it's enough to find two disjoint $C_5$ extensions of $\mathbb{Q}_5$ and take their compositum. $C_5$ extensions don't seem too easy to construct, so by the local Kronecker-Weber theorem one could just look for cyclotomic extensions of degree divisible by $5$ and try to identify a subextension of degree $5$.
I've tried to find a good source about cyclotomic extensions of $p$-adics, but this seems to be a topic missing from all field theory books. Books on algebraic number theory seem to omit this topic too and our graduate algebra class didn't cover it either.
Now my intuition would guide me as follows. Working in $\mathbb{Q}$ one would immediately find $\mathbb{Q}(\zeta_{11}+\zeta_{11}^{-1})$ as a cyclic degree $5$ extension. We also have that $[\mathbb{Q}(\zeta_{25}):\mathbb{Q}]=20$, so that $[\mathbb{Q}(\zeta_{25}+\zeta_{25}^{-1}):\mathbb{Q}]=10$ and with $(\zeta_{25}+\zeta_{25}^{-1})^2=\zeta_{25}^2+2+\zeta_{25}^{-2}$ one could expect that setting $\alpha=\zeta_{25}^2+\zeta_{25}^{-2}$, we would have
$$[\mathbb{Q}(\zeta_{25}+\zeta_{25}^{-1}):\mathbb{Q}(\alpha)]=2\Rightarrow [\mathbb{Q}(\alpha):\mathbb{Q}]=5$$
(Note, I haven't actually checked if this first extension has degree $2$)
Finally, $\mathbb{Q}(\zeta_{25})\cap \mathbb{Q}(\zeta_{11})=\mathbb{Q}$, so we would have that
$$\mathbb{Q}(\zeta_{11}+\zeta_{11}^{-1},\alpha)/\mathbb{Q}$$
is a $C_5\times C_5$ extension of $\mathbb{Q}$. My question is then that does this work if we replace $\mathbb{Q}$ with $\mathbb{Q}_5$? More specifically:

What would the Euler totient functions look like if defined as the order of the $n^\textrm{th}$ cyclotomic extension of $\mathbb{Q}_p$? If $p-1\mid n$, it would at least have to be different. Is $[\mathbb{Q}_5(\zeta_{25}):\mathbb{Q}_5]=[\mathbb{Q}(\zeta_{25}):\mathbb{Q}]$?
Can we still expect to have e.g. $\mathbb{Q}_p(\zeta_n)\cap \mathbb{Q}_p(\zeta_m)=\mathbb{Q}_p(\zeta_{\gcd(n,m)})$? What about other local fields of characteristic $0$?

I haven't thought about these really at all as I would rather just find a good reference for this topic.

REPLY [2 votes]: Since you allude to the local version of the Kronecker-Weber theorem, let us exploit it to the core. The maximal abelian extension of $Q_p$  is generated by all roots of unity. It is the compositum of two specific extensions : $Q_{p,nr}$ , generated by all roots of unity of order prime to p, which is also the maximal unramified extension of $Q_p$ ; and $Q_{p,cyc}$, generated by all the p-power roots of unity, which is totally ramified. To obtain an abelian extension of type (p,p) of $Q_p$, just combine the two cyclic extensions of degree p contained in $Q_{p,nr}$ and $Q_{p,cyc}$ (these are easy to obtain, especially in your particular case p = 5; this is essentially  Ted's answer). Note that by local class field theory (¤), the Galois group over $Q_p$  of its maximal abelian extension of type (p, p,…, p) is isomorphic to $(Q_p)*$/p-powers , so that in your example p = 5 , this maximal extension is just the field you are looking for.
(¤) Your question having been posted 4 years ago, you must have learnt CFT by now. See specifically  Serre's chapter 6, $3.1 in Cassels - Fröhlich's "Algebraic Number Theory" .
EDIT   Actually, as a Z/pZ vector space, it can be shown that $(Q_p)*$/p-powers has dimension 2, so that your example p = 5 is not a particular case .