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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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up vote 5 down vote accepted

Your question was asked and answered on MO before. The examples you give are essentially it, in the sense that the field has to be real closed and the algebraic closure is obtained by adding a square root of -1. This is the Artin-Schreier theorem.

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Thanks, I'd never heard of MO before. –  John Smith Dec 11 '10 at 4:37

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