I am asking a question that I should feel ashamed of!
Suppose $K$ is a number field and $K/\mathbb{Q}$ is a finite Galois extension of rank $n$, say \begin{equation} K=Q[x]/f(x) \end{equation} where the degree of the polynomial $f(x)$ is $n$. Let's consider $X:=\text{Spec}\,K$ to be a $\text{Spec}\,\mathbb{Q}$-variety. The variety $X_{\overline{\mathbb{Q}}}$ is just, \begin{equation} X_{\overline{\mathbb{Q}}}:=\text{Spec}\,K \otimes_{\mathbb{Q}}\overline{\mathbb{Q}}=\text{Spec}\,\overline{\mathbb{Q}}[x]/f(x) \end{equation} which is a variety defined over $\overline{\mathbb{Q}}$ consists of $n$ points that correspond to the $n$ roots of $f(x)$.
Question 1: The Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ has a right action on the variety $X_{\overline{\mathbb{Q}}}$, what is this action? I guess its action on the $n$ points is like the Galois group $\text{Gal}(K/\mathbb{Q})$ permutes the $n$ roots?
Now the etale cohomology group \begin{equation} H^0_{et}(X_{\overline{\mathbb{Q}}},\mathbb{Q}_\ell) \end{equation} is a representation of the Galois group $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
Question 2 This group is an $n$ dimensional $\mathbb{Q}_\ell$ vector space, I feel it is just the representation that factors through $\text{Gal}(K/\mathbb{Q})$, but I could not make this precise, could anyone give a rigourous construction?