Remarks

A morphism of fields is nothing but a morphism of the underlying rings. Observe that such a morphism takes x^-1 to the inverse of the image of x.

In Part 3, the morphism f : K -> K need not be surjective. For instance, let K be the rational functions field Z/2Z(X). Then X is not in the image of x -> x².

If K is finite, then, by the pigeon hole principle, the morphism x -> x^p is surjective and hence an isomorphism.

In fact, the fixed points of a morphism R -> R of rings also form a subring of R.