I am currently studying the theory of field extensions.
One of the exercises in my book is causing me a trouble. The problem is:
Give an example of a field $E$ containing a proper subfield $K$ such that $E$ is embeddable in $K$ and $[E:K]$ is finite.
First, I could't find any example. To me, it seems like any field in typical textbook cannot be an example.
Second, it seems to me that the problem states that it is possible for two fields to be a field extension of the other but they are not the same(since $K$ is proper). Is it correct? I thought that field extension can be used to establish some partial order, if we restrict fields that are subfields of some given field (for example, algebraically closed field of $E$, in this case).
Any examples or hints would be highly appreciated.