Continuous automorphisms of $\mathbb{C}_p$

Question

Goodmorning my queston is: Every continuous automoprhism of $\mathbb{C}_p$ is on the form

$$\sigma(x)=\lim_{n\to \infty}\bar{\sigma}{(x_n})$$

where $x =\lim_{n\to \infty}x_n $ and $\bar{\sigma}\in Gal(\bar{\mathbb{Q}}_p/ \mathbb{Q}_p)$ and $x_n \in \bar{\mathbb{Q}}_p$ ? Beacause if $\sigma$ is on this form is surely an automorphism of $\mathbb{C}_p$ but the converse is true? I think that is not true but...

Thank you for the answers!

Answer 1

It’s not clear to me whether you are asking, as part of your question, whether every automorphism of $\Bbb C_p$ must be continuous.

I think that this is the nub, the crux, of the matter: if you start with any automorphism of $\Bbb C_p$, then it restricts to a continuous automorphism $\sigma_0$ of $\bar{\Bbb Q}_p$. Since $\sigma_0\in\text{Gal}(\bar{\Bbb Q}_p/\Bbb Q_p)$, it extends uniquely to the closure, i.e. to $\Bbb C_p$, and this extension must be the original $\sigma$, if this $\sigma$ is known to be continuous.

But: whether every automorphism of $\Bbb C_p$ is continuous (surely true?), this seems to me to be a more delicate question, which I’m not prepared to attack.