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Let $E$ be an algebraic extension of a field $F$ and let $\sigma: F\to L$ be an embedding of $F$ in an algebraically closed field $L$, where $L$ is the algebraic closure of $\sigma F$.

Does there exist at least an extension of $\sigma$ to an embedding of $E$ in $L$?

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Yes.

Consider the set of all $\sigma'\colon E'\to L$ where $F\le E'\le E$ and $\sigma'|_F=\sigma$. We can apply Zorn's lemma to find a maximal $\sigma'\colon E'\to L$. Suppose $E'\ne E$ and let $\alpha\in E\setminus E'$ and $f\in E'[X]$ its irreducible polynomial. Then $f$ has no roots in $E'$, hence $\sigma'(f)$ has no roots in $\sigma(E')$, but it does have a root $\beta\in L$.We can extend $\sigma'$ to $E'[\alpha]$ by mapping $\alpha\mapsto \beta$ (i.e., this is well-defined). As this contradicts maximality, we must have $E'=E$.

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