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What is the condition for a field to make the degree of its algebraic closure over it infinite?

More generally, what is known about subfields $\mathbb{F}\subset \mathbb{C}$ such that the degree $[\mathbb{C}:\mathbb{F}]$ is finite? We all know, for example, that $[\mathbb{C}:\mathbb{C}]=1$ and $[\mathbb{C}:\mathbb{R}]=2$ (or we should all know this). However, $\mathbb{C}$ has many subfields. Can you construct/identify one with $[\mathbb{C}:\mathbb{F}]=3$?

This is a question which I think about whenever I teach abstract algebra.

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    Those are the only possibilities. See http://math.stackexchange.com/questions/23729/what-is-the-condition-for-a-field-to-make-the-degree-of-its-algebraic-closure-ove .2011-03-09

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