Suppose $F$ is a field and $R$ is a ring. The function $f\colon F\to R$ is a surjective homomorphism. Prove that $R$ is either the trivial ring, or $R$ is isomorphic to $F$.
We have a ring homomorphism $f : F \to R$ here are the key facts:
Take the inverse image of the kernel, if it's $(0)$ the rings are isomorphic. If it's $F$ the homomorphism is trivial.