Every Field Homomorphism Is Injective

Proof

Let $\varphi :\mathbb{K}\to \mathbb{L}$ be a field homomorphism. This means $\varphi$ is also a ring homomorphism. This means that the kernel of $\varphi$ is an ideal of $\mathbb{K}$, that is $\ker (\varphi )⊴\mathbb{K}$.

Since $\mathbb{K}$ is a field, the only ideals are the zero ideal and the ring itself.

If $\ker (\varphi )=\mathbb{K}$, then $\varphi (a)=0\mathrm{\forall }a\in \mathbb{R}$. This implies that $\varphi (1)=0$, which contradicts the fact that $\varphi$ is a homomorphism.

Hence $\ker (\varphi )=\{0\}$, and $\varphi$ is injective .