Homomorphism in Rings

Question

I have the following true/false statement from a worked example:

If $F_1, F_2$ are fields and $\phi: F_1 \rightarrow F_2$ is a homomorphism, then $\phi$ is either the zero map or an isomorphism.

The solution says that this is False and gives the identity map $\mathbb{R} \rightarrow \mathbb{C}$ as a counterexample.

I am a bit confused by this. Obviously the identity map is not the zero map. When I try to check the conditions for isomoporphism I get confused on the domain\range over which the identity map is defined.
Can someone explain why this example disproves the statement?
Thank you in advance for any answers/comments.

Answer 1

So $\mathbb{R}$ is a subset of $\mathbb{C}$ so the identity map is clearly a homomorphism from $\mathbb{R}$ to $\mathbb{C}.$ As you said, it is clearly not the zero map but the map is not surjective! What maps to $i?$ Hence, it cannot be an isomorphism as it is not bijective.

Answer 2

The identity map, by definition is a function $i : \mathbb{C} \to \mathbb{C}$. With a little abuse of language we refer to the identity map of $\mathbb{R}$ in $\mathbb{C}$ as the restriction of the above function to $\mathbb{R}$, hence the morphism you are actually considering is $$ i_{| \mathbb{R}} : \mathbb{R} \to \mathbb{C} $$

which is both non zero and non surjective