Let $a\in\overline{ \Bbb Q_p}$, $\sigma\in Gal$( $\overline{ \Bbb Q_p}/ \Bbb{Q}_p$）, then, why $|a|=|\sigma (a)|$, where $|\cdot|$ denotes absolute value on $\overline{ \Bbb Q_p}$?
I think we should prove $\operatorname{ord}_\pi(a)$ and $\operatorname{ord}_\pi(\sigma(a))$ is the same, but I cannot go further. Thank you in advance.
The reason is that absolute value $|-|$ on $\overline{\Bbb Q_p}$ is the unique absolute value that is an extension of the p-adic absolute value on $\Bbb Q_p$ (it's enough to prove this for finite extensions and there it is a standard fact).
If $\sigma \in \mathrm{Gal}(\overline{\Bbb Q_p}/\Bbb Q_p)$, then $a\mapsto |\sigma(a)|$ is another absolute value that extends the p-adic absolute value on $\Bbb Q_p$. Hence it is equal to the standard absolute value.