is it always true that $F'|F$ is a normal extension?$F'$ means the algebraic closure of $F$.
what conditions are necessary for that?
thanks
2026-04-02 17:41:07.1775151667
normality of algebraic closure
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So one of the properties of a normal extension $L$ of $K$ is that every embedding $\sigma$ of $L$ in $K'$ (the algebraic closure of $K$) that restricts to the identity on $K$, satisfies $\sigma(L) = L$. So your question reduces to if every embedding $\sigma$ (which fixes $K$) that takes $K'$ into $K'$ also has $\sigma(K') = K'$.
What is really more interesting is the question of normal extensions that sit in between $K$ and $K'$.