Suppose $F$ is a finite extension of $\mathbb Q_l$ for some prime $l$, then we know the absolute value on $\mathbb Q_l$ can be uniquely extended to $F$. Hence $F$ is equipped with the metric topology.
On the other hand, $F$ is a finite dimensional vector space of $\mathbb Q_l$(say, $F\simeq \mathbb Q_l^{\oplus n}$) , then it can be equipped with the product topology.
Are these two topologies the same?