Let $A$ be an abelian variety over a $p$-adic field $K$. If $K(A_{p^\infty})$ is the field extension of $K$ obtained by adjoining the coordinates of all $p$-power division points of $A$. By the Weil pairing, it is known that $K(A_{p^\infty})$ contains the field $K(\mu_{p^\infty})$, the field obtained by adjoining to $K$ all the $p$-power roots of unity.

Now, let $X$ be a proper smooth variety over $K$ and consider the $p$-adic Galois representation $$ \rho : G_K \rightarrow GL(V),$$ where $G_K:=\text{Gal}(\bar{K}/K)$ and $V = H^i_{et} (X_{\bar{K}}, \mathbb{Q}_p)$. Let $F := {\bar{K}}^{\text{ker}\rho}$ be the fixed subfield of $\bar{K}$ by the kernel of $\rho$.

Question 1: Does $F$ contain $K(\mu_{p^\infty})$ in this general case?

By comparing with the abelian variety case, I think that for the answer to Question 1 to be yes, one needs to generalize the Weil pairing.

Question 2: Is there a "generalized Weil pairing" for proper smooth varieties $X$ given above?