Let $L/K$ be a field extension. Is $L$ a subfield of some algebraic closure $\bar K$ of $K$?
2026-03-27 13:20:17.1774617617
If $L/K$ is algebraic, is $L$ a subfield of some algebraic closure $\bar K$ of $K$?
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Take an algebraic closure $F$ of $L$. I contend that $F$ is an algebraic closure of $K$.
Indeed, it is algebraically closed by assumption; if $x\in F$, then $x$ is algebraic over $L$, hence also over $K$.