Every Field Homomorphism Is Injective

Proof

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

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

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

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