I have a question about embedments regarding field theory.
If we have two groups $G$ and $H$ and a group homomorphism $\varphi: G\to H$ that is injective, we can identify $G$ with a subgroup of $H$ by the homomorphism theorem.
Now, if we have a field $K$ and two field extensions $L$, $L'$ of $K$ such that there is a $K$-Homomorphism $\varphi: L\to L'$. We know that $\varphi$ always is injective as a homomorphism of fields. Does this mean that $L$ is a subfield of $L'$? Are images of fields even subfields?
The first part: Actually, it could be a subfield up to isomorphism. Take as an example a field extension E of a field K. If there is a transcendental element a in E, then K[a] and K[T] are isomorphic as rings. Hence you could take the quotient K(T):=Q(K[T]) which is also a field extension of K. Since K[a] and K[T] are isomorphic, as there is the inclusion from K[T] into K(T), we have an injective map from K[a] into K(T) but K[a] is not esentially in K(T) and hence it's not a subfield. Since $\varphi$ is injective, by th first isomorphism theorem, L $ \cong $ Im($\varphi$) and hence the image is a subfield of L'.