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Pete L. Clark
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Algebraicity of the completion of a field

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.

I certainly plan to think about this myself, but since MO is a site for interesting math questions, I felt that others should be able to share in the fun.

Pete L. Clark
  • 65.4k
  • 12
  • 241
  • 381