If $E$ is a finite algebraic extension of the field $F$, then we can find a normal closure of $E$ over $F$. What can we say if the extension $E$ is not finite?
2026-04-02 14:19:11.1775139551
Normal closures of transcendental extensions
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Take the algebraic closure $\overline{E}$ of $E$. This is an algebraically closed field which contains $F$, so it must contain $\overline{F}$. Therefore $\overline{E}$ is a normal extension of $F$ which contains $E$. So the family of all the normal extensions of $F$ which contain $E$ is nonempty. Now take the intersection of all these guys: you will get the normal closure of $E$ over $F$.