Field homomorphisms are injective

Suppose $𝐹, 𝐾$ are fields and $𝜑 : 𝐹 \to 𝐾$ is a ring homomorphism. Then $𝜑$ is injective, and is thus a field extension. #m/thm/ring

Proof

Note $k e r 𝜑$ is necessarily a proper ideal of $𝐹$ , and $𝑅$ is a field iff it has no nonzero proper ideals, thus $k e r 𝜑 = 0$ .

#state/tidy | #lang/en | #SemBr