Vtyjkyuk

The following is Exercise V.3.21 from Hungerford's Algebra (GTM 73).

Let $F$ be algebraic over $K$. $F$ is normal over $K$ if and only if for every $K$-monomorphism of fields $sigma:Fto N$, where $N$ is any normal extension of $K$ containing $F$, $sigma(F)=F$ so that $sigma$ is a $K$-automorphism of $F$. [Hint: Adapt the proof of Theorem
3.14, using Theorem 3.16.]

Theorem 3.14 is the usual characterization of normal extensions. Theorem 3.16 states the properties of normal closures. However, I don't think we need normal closures here.

We shall prove that the above condition implies the following, which will prove that $F$ is normal over $K$: If $overlineK$ is any algebraic closure of $K$ containing $F$, then for any $K$-monomorphism of fields $sigma:FtooverlineK$, $operatornameimsigma=F$ so that $sigma$ is actually a $K$-automorphism of $F$.

Proof. Extend $sigma:FtooverlineK$ to a $K$-monomorphism $overlinesigma:NtooverlineK$ (this is possible since $N$ is algebraic over $F$). Since $N$ is normal over $K$, we have $operatornameimoverlinesigma=N$, so $sigma$ actually maps $F$ into $N$. By hypothesis $operatornameimsigma=F$, so we're done.

The other direction is rather obvious, and does not need normal closures either. So I have not made any use of normal closures here. But why is the hint asking me to do so? Is my proof correct?

Everything you say seems completely correct, but I think our problem is that the exercise is sloppily stated.

I think that the exercise should be stated as follows: Let $Fsupset K$ be an algebraic extension and $N$ a normal extension of $K$ containing $F$. Then $F$ is normal over $K$ if and only if for every $K$-morphism $sigma:Fto N$, $sigma(F)=F$.

What he seems to want is that to check for normality of the extension $Fsupset K$, you need only embed the situation in some (perhaps well-chosen) normal extension of $K$, not (for instance) an algebraically closed field containing $K$. Then this would be a useful theorem.

Notice that our author has violated the principle of putting all the quantifiers at the beginning. Let this be a lesson to all of us: you’re asking for trouble if you embed quantifiers in the middle of a proposition.