definition of an isomorphism between field extensions

Question

I am looking at this definition: https://gyazo.com/9fd014a74496a37356c51aced862b69b

I am having troubles understanding the bottom paragraph, specifically what it means "if we identify $K$ and $i(K)$ and $L$ and $j(L)$ then $i$ and $j$ are inclusions" what does this mean exactly?

Answer 1

A morphism of field extensions is either injective or the zero map. This can be seen by considering the kernel of such a morphism. It is an ideal of the domain, which is a field, and therefore, must either be zero or the entire field. So if you identify the image with another field, then that rules out the case of the zero map.