Completion of a metric space vs. field extension

Question

Given a field $F$ and a metric $\mu$, is the completion of $F$ with respect to $\mu$ always a field? Additionally, is there an algebraic field extension of $F$ that is isomorphic to the completion of $F$ by $\mu$?

Answer 1

Jack M answered the question in the comments above. I am posting this answer just so that I can close out the question.

Answer:

Jack M wrote:

Of course not, because the metric need not have anything to do with the field. As far as the metric is concerned, $F$ is just a set. Things might get interesting if the field operations are required to be continuous for the metric.

For point (2), note that $\Bbb{R}$ is not an algebraic extension of $\Bbb{Q}$