Definition 9.26.9. Let $K/k$ be a field extension.

1. The algebraic closure of $k$ in $K$ is the subfield $k'$ of $K$ consisting of elements of $K$ which are algebraic over $k$.

2. We say $k$ is algebraically closed in $K$ if every element of $K$ which is algebraic over $k$ is contained in $k$.

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