Definition 9.10.3. Let $F$ be a field. We say $F$ is algebraically closed if every algebraic extension $E/F$ is trivial, i.e., $E = F$. An algebraic closure of $F$ is a field $\overline{F}$ containing $F$ such that:

1. $\overline{F}$ is algebraic over $F$.

2. $\overline{F}$ is algebraically closed.

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