Let $(k,|.|)$ be a complete ultrametric field, and let $F$ be the completion of the algebraic closure $k^{alg}$ with respect to $|.|$. Let G be the absolute Galois group of $k$. I denote by $\mathbb{A}_k^{1,an}$ the analytic Berkovich line over $k$, and let $p:\mathbb{A}_F^{1,an}\to \mathbb{A}_k^{1,an}$ be the canonical projection.
The goal of my question is to understand the nature of the point $p(\pi)$ where $\pi\in F$ is a transcendental element over $k$.
Given an element $a\in F$, if for all $\sigma\in G$ we have $\sigma(a)=a$, do we have $a\in k$? or can we find a transcendental element $\pi$ such that for all $\sigma\in G$, $\sigma(\pi)=\pi$?