At the end of my 8410 class today (see http://www.math.uga.edu/~pete/MATH8410.html if you care), one of my students asked me the following very interesting question:

Let $(K,|\ |)$ be a normed field, with completion $(\hat{K},| \ |)$. Suppose $\hat{K}$ is algebraic over $K$. Must we then have $\hat{K} = K$?

As I have mentioned here before, I feel very lucky to be getting such penetrating questions. This one I was not able to answer on the spot, although I remarked that it is true in all of the most familiar examples and that the (possible) lack of algebraicity of the completion is a key motivation for considering the Henselization instead.

Edit: the answer is no, as I have just heard from one of my students. I have encouraged him to come to this site and register the answer.

To make the question more interesting, suppose we ask whether $\hat{K}/K$ can be finite and nontrivial?