# Field extension question

A question I recall from Carl Linderholm's "Mathematics Made Difficult", Chapter 3 Exercise 8.

A farmer acquires and algebraically closed field by extending his field finitely.
What can be said about the original field?


What can be said? I immediately see two fields with this property, ${\Bbb R}$, and $\overline{\Bbb Q}\cap {\Bbb R}$. Is this it? or is there a better characterization? Do these fields have to contain ${\Bbb Q}$?

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