Construct an embedding

Question

I'm dealing with this problem from the book "Field Theory" (Steven Roman)

Suppose $F$ and $E$ are fields and $\sigma : F \rightarrow E $ is an embedding. Construct an extension of $F$ that is isomorphic to $E$.

How can we extend?

Answer 1

It may probably be something about the extension of embeddings. Namely, if you extend $\sigma$ to $\sigma$$^-$ you will get the extension

$\sigma $$(F)$$\lt $ $\sigma$$^-$$(E)$

And note that $\sigma$ is a monomorphism and you will get $E$ isomorphic to $\sigma$$^-$$(E)$.

However, it may not be that much easy, needs some comment.

Answer 2

Then all I can say about $F'$ is that it may be exactly $(\sigma^-)^{-1} (E)$, which definitely is isomorphic to E.