Say $F$ and $K$ are fields and there is a homomorphism $\phi : F \to K$. Is it required that $\phi(0) = 0$?
I think it's true, since $\phi(0 \cdot x) = \phi(0) \cdot \phi(x) \implies \phi(0) = \phi(0) \cdot \phi(x)$ is only true when either $\phi(0) = 0$ or $\phi(x) = 0$. If $x \neq 0$, the equality still has to hold true so $\phi(0) = 0$.
Or similarly, $\phi(0 + x) = \phi(0) + \phi(x) \implies \phi(x) = \phi(0) + \phi(x) \implies \phi(0) = 0$.
$\phi(0)=0$ is true for every ring homomorphism $\phi:R\to S$, and fields are special kind of rings.
It follows directly from $$\phi(0)=\phi(0+0)=\phi(0)+\phi(0).$$