Serre's Modularity Conjecture -- Weight I was reading Serre's paper "Sur les Représentations Modulaires de Degré $2$ de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). 
Following his notation, let $G_p$ be the absolute Galois group of $\mathbb{Q}_p$, $I$ the (absolute) inertia group and $I_p$ the wild inertia group.
In order to define the weight $k$ attached to a Galois representation it is only needed to look locally at $p$, so consider a continuous representation 
\begin{equation*}
\rho_p:G_p\longrightarrow \mathbf{GL}_2(\bar{\mathbb{F}}_p)
\end{equation*}
Now the paper splits into several cases. The one I am looking at is in page $186$ (of the Duke Mathematical Journal where it appeared) and corresponds to the case where $I_p$ does not act trivially and the restriction of the representation to $I$ is given by
\begin{equation*}
\rho_p|I=\begin{pmatrix}\chi^{\alpha+1} & *\\ 0 & \chi^{\alpha}\end{pmatrix}
\end{equation*}
for some $\alpha\in\{0,\ldots,p-2\}$, where $\chi$ is the cyclotomic character. It is clear that $\rho_p(I)$ is the Galois group of some totally ramified extension $K$ of $\mathbb{Q}_p^{nr}$ and that $\rho_p(I_p)$ is the Galois group of $K/K^{tr}$, where $K^{tr}$ is the maximal tamely ramified extension of $\mathbb{Q}_p^{nr}$ contained in $K$.
My question is the following: a few lines down he asserts that by Kummer theory we can deduce that $K=K^{tr}(x_1^{1/p},\ldots,x_m^{1/p})$, with $x_i\in\mathbb{Q}_p^{nr}$. I can see, by Kummer theory, that $K$ is of this form for $x_i\in K^{tr}$. Why is it true that we can place these $x_i$ in $\mathbb{Q}_p^{nr}$?
Thank you for your answers!
 A: The conjugation of an upper triangular matrix 
$\begin{pmatrix} a & b \\ 0 & d \end{pmatrix}$ 
takes another upper triangular matrix
$\begin{pmatrix} \alpha & \beta \\ 0 & \delta \end{pmatrix}$ 
to $\begin{pmatrix} \alpha & \frac{a}{d} \beta \\ 0 & \delta \end{pmatrix}$.
So, for the representation $\rho_p$ in question, 
inertia acts on the upper right-hand `*' via the ratio of the two characters,
which is $\chi,$ the mod $p$ cyclotomic character. You can check that
the Kummer theoretic statement you have already used to get the $x_i$
in $K^{tr}$ can be refined, using this additional information,
to show that the $x_i$ are in $\mathbb Q_p^{nr}$.
A: One needs the following facts that Serre observes in the given situation:


*

*$K^{\mathrm{tr}} = \mathbb{Q}^{\mathrm{nr}}_p(\mu_p)$ where $\mu_p$ are the $p$-roots of unity;

*the extension $K/\mathbb{Q}^{\mathrm{nr}}_p(\mu_p)$ is $p$-elementary abelian;

*the action of $\mathrm{Gal}(\mathbb{Q}^{\mathrm{nr}}(\mu_p)/\mathbb{Q}^{\mathrm{nr}}_p) \cong (\mathbb{Z}/p\mathbb{Z})^\times$ on $\mathrm{Gal}(K/\mathbb{Q}^{\mathrm{nr}}_p(\mu_p))$ by conjugation is "the obvious one", i.e. by multiplication.


Abstracting a bit, let us consider the following general situation: Let $K$ be a field, $p \neq \mathrm{char}(K)$ a prime number, $K(\mu_p)$ the field obtained by adjoining the $p$-th roots of unity, and let $L/K(\mu_p)$ be a $p$-elementary abelian extension such that $L$ is also Galois over $K$. We have the cyclotomic character $\chi: \mathrm{Gal}(K(\mu_p)/K) \hookrightarrow (\mathbb{Z}/p\mathbb{Z})^\times$ defined by $\sigma(\zeta) = \zeta^{\chi(\sigma)}$ for $\zeta \in \mu_p$. We have also a conjugation action of $\mathrm{Gal}(K(\mu_p)/K)$ on $\mathrm{Gal}(L/K(\mu_p))$. The fact from Kummer theory that Serre uses is:

Theorem: $\mathrm{Gal}(K(\mu_p)/K)$ acts on $\mathrm{Gal}(L/K(\mu_p))$ as multiplication via the cyclotomic character if and only if $L = K(\mu_p)(x_1^{1/p},\ldots,x_m^{1/p})$ for certain $x_i \in K^\times$ and $m \in \mathbb{N}_0$.

To see this, let $\Delta_L := (K(\mu_p)^\times \cap {L^\times}^p)/{K(\mu_p)^\times}^p$, so that $L = K(\mu_p)(\Delta_L^{1/p})$. We have the perfect Kummer pairing
$$ \mathrm{Gal}(L/K(\mu_p)) \times \Delta_L \longrightarrow \mu_p,$$
which is $\mathrm{Gal}(K(\mu_p)/K)$-equivariant. Since $\mathrm{Gal}(K(\mu_p)/K)$ acts on $\mu_p$ via the cyclotomic character, it follows that the action on $\mathrm{Gal}(L/K(\mu_p))$ is also via the cyclotomic character if and only if it is trivial on $\Delta_L$. The claim then follows from
$$ \frac{K^\times}{{K^\times}^p} = \Bigl(\frac{K(\mu_p)^\times}{{K(\mu_p)^\times}^p}\Bigr)^{\mathrm{Gal}(K(\mu_p)/K)},$$
which is the isomorphism $\mathrm{H}^1(K, \mu_p) \cong \mathrm{H}^1(K(\mu_p), \mu_p)^{\mathrm{Gal}(K(\mu_p)/K)}$ that one gets from the inflation-restriction sequence.
