Serre's Modularity Conjecture -- Weight

Question

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!

Answer 1

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 $\mathbb Q_p^{nr}$.