embedding of a finite algebraic extension

Question

In one of my courses we are proving something (so far, not surprising) and using the fact: if $F$ is a finite algebraic field extension of $K$, there is an embedding of $F$ into $K$. Well, doesn't seems to me that we can really embed $F$ into $K$, since $F$ is bigger, but can we at least prove there is a homomorphism from $F$ to $K$?

Answer 1

Any homomorphism of fields must be zero or an embedding as there are no nontrivial ideals of any field. There is always the natural inclusion $i: K\rightarrow F$ if $K\subseteq F$, but rarely do we have an embedding $F \rightarrow K$.

For a simple example, there is no embedding $\Bbb C\rightarrow \Bbb R$, as only one has a root of $x^2+1$ and an embedding will preserve roots of this polynomial. There are in fact examples of algebraic extensions $K\subseteq F$, with embeddings $F\rightarrow K$ (i.e. $K(x)\rightarrow K(x^p)$) .