Lemma 15.14.5. Let $A$ be a normal domain. Then $A$ is absolutely integrally closed if and only if its fraction field is algebraically closed.
Proof. Observe that a field is algebraically closed if and only if it is absolutely integrally closed as a ring. Hence the lemma follows from Lemmas 15.14.3 and 15.14.4. $\square$
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