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

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 $$L$$?

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 $$E'=E$$.