Field Homomorphism

Note that this is simply an extension of the definition of a ring homomorphism for a ring with identity.

It can also be thought of as a generalisation of a group homomorphism over the two underlying groups on which the field is constructed. As with the definition for a ring homomorphism, we don't need to explicitly require $\varphi (0_{\mathbb{F}})=0_{\mathbb{E}}$, since this follows from the first axiom above. The second is however insufficient in deriving the third as it alone doesn't exclude the constant zero function.

It can easily be proven that every field homomorphism is injective.