What are the algebraically closed subfields of $\mathbb{C}$ ?
There is $\mathbb{C}$, there is $\bar{\mathbb{Q}}$... but what else ?
2026-04-22 21:07:30.1776892050
Algebraically closed subfields of $\mathbb{C}$
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Since $\Bbb C$ has uncountable transcendance degree over $\Bbb Q$, there is plenty of room between $\overline{\Bbb Q}$ and $\Bbb C$. Just adjoin any countable collection of algebraically independent complex numbers to $\Bbb Q$ and take the algebraic closure. And some uncountable collections will fail to generate all of $\Bbb C$ as well. There are really too much of these for any serious attempt to classify them.