Question about embeddings in algebraically closed fields I was looking at the two following well known propositions:
$1$) Let $E|F$ be a finite separable extension of degree n, and let $\sigma$ be an embedding of $F$ in $C$, where $C$ is an algebraic closure of $E$. Then $\sigma$ extends to exactly $n$ embeddings of $E$ in $C$;
$2$) The extension $E|F$ is normal if and only if every $F$-monomorphism of $E$ into an algebraic closure $C$ is actually an $F$-automorphism of $E$.

Observing the proofs, it seems to me that actually instead of $C$ we can pick any algebraically closed field $L$ containing $F$ (we don't need to assume that it contains also $E$ in these propositions, right?)  $\textbf{in both propositions}$. 

Is it correct? (I'm not so sure for prop 2)). What I'm thinking of, is the typical situation in number theory where you have $\mathbb C$ instead of $C$. In general $\mathbb C$ is not the algebraic closure of $E$, but just an algebraically closed field. So why in many texts the authors put algebraic closure instead of simply algebraically closed field?

Because algebraic closures are intrinsic in terms of the field, and any algebraically closed field usually means that there's some topological structure. Eg. with $\Bbb C$ vs $\overline{\Bbb Q}$ the latter is purely algebraic in terms of the polynomials over $\Bbb Q$, but if you want to prove a result about embeddings like this one you prefer into $\overline{F}$ rather than $L\supseteq\overline{F}$. Because often there are many choices for such an $L$, and you don't want to write a separate proof for all cases. I would not want to prove this result two different times one for eg. $\Bbb C$ and another for $\overline{\Bbb Q_2}$. And if I'm dealing with a field of positive characteristic, obviously something like $\Bbb C$ is not even available to me.
