Example of non isomorphic field extensions

Question

What is an example of non isomorphic field extensions? I would like such a non example as I think it would explain the significance of the commutativity of the square condition in an isomorphism of field extensions $L/K\cong M/N$ where $\mu$, $\sigma$ are isomorphisms:

Answer 1

Note that it is well known that $\mathbb{C}$ contains a copy of itself as a proper subfield. In such situation you can take $K=L=M=\mathbb{C}$ and $N$ a proper subfield of $\mathbb{C}=M$ isomorphic to $\mathbb{C}$.

Of course those fields are pairwise isomorphic, but no extension isomorphism (i.e. the diagram) exists. Because extension isomorphism also preserves degree. And one extension is of degree $1$ and the other of (necessarily) infinite degree.

And so extension isomorphism is more than just pair of isomorphisms.

Answer 2

Here's another example where the distinction is often practically important. If $K$ is a field of characteristic $p$, then $K^{1/p}$ ($p$-th roots taken in an algebraic closure) is abstractly isomorphic to $K$ via the $p$-th power Frobenius map $K^{1/p} \to K$. But if $K$ is not perfect then the inclusion $K \hookrightarrow K^{1/p}$ is not an isomorphism.