I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.

If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that would be impossible. But $\mathbb C_p$ is not spherically complete, so we can apply Hahn-Banach. An d my problem is still opened....

Does anyone have a hint or an example of such automorphism? Thanks in advance

I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.

If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that would be impossible. But $\mathbb C_p$ is not spherically complete, so we can not apply Hahn-Banach. And my problem is still opened....

Does anyone have a hint or an example of such automorphism? Thanks in advance