A question based on algebraically closed field and extending an isomorphism to monomorphism over different fields

Question

The following question was asked in mind term exam (held in november) and I was unable to solve it. Now I was trying to solve it again and failed this time too.

So, I am in need of help.

F is an algebraic closure of K if and only if F is algebraic over K and for every algebraic field extension E of another field $K_1$ and isomorphism of fields $\sigma : K_1 \to K , \sigma$ extends to a monomorphism $E \to F$.

If given is that F is an algebraic closure of K then it is clear that F is algebraic over K but I am not clear on how should I prove that $\sigma$ extends to a monomorphism $E\to F$. Actually I am bad at proving results when any type of morphism is to be extended.

For the converse, I am not able to get any idea.

It is my humble request to you to guide me in this.

Answer 1

The part about extending $\sigma$ is a bit lengthy and usually involves two steps:

1. Proving the case where $E=K(\alpha)$ is a simple extension field
2. Concluding the general case by Zorn's lemma.

This should be covered in pretty much any algebra/field theory book.
For the other direction. Take an algebraic closure $C$ of $F$. By applying the hypothesis to $\sigma=\operatorname{id}_K:K\to K$ we get a $K$-homomorphism $f:C\to F$. The image $f(C)$ will then contain $K$ and is still algebraically closed. Since $F/F(C)/K$ is algebraic it follows that $F=F(C)$, hence $F$ is algebraically closed.

(Also note that a homomorphism of fields is automatically a monomorphism)