# Is any homomorphism between two isomorphic fields an isomorphism?

Is any homomorphism between two isomorphic fields an isomorphism? What I mean is that two fields are called isomorphic if there exist one homomorphism between them . But not sure if existence of one isomorphism means there exist no non-bijective homomorphism between . An explanation or a counter example would help . Please.

You mean if there exists an isomorphism between them. It is not true that a homomorphism between two isomorphic fields needs to be an isomorphism. For example, the natural inclusion $\mathbb{C} \to \overline{ \mathbb{C}(t) }$ is not an isomorphism, but the two fields in question are isomorphic. Perhaps a simpler example is the inclusion $\mathbb{C}(x_2, x_3, x_4, ...) \to \mathbb{C}(x_1, x_2, x_3, ...)$.