# Field Homomorphism is either Trivial or Injection/Proof 1

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## Theorem

Let $\struct {E, +_E, \times_E}$ and $\struct {F, +_F, \times_F}$ be fields.

Let $\phi: E \to F$ be a (field) homomorphism.

Then $\phi$ is either an injection or the trivial homomorphism.

## Proof

This is an instance of Ring Homomorphism from Field is Monomorphism or Zero Homomorphism.

$\blacksquare$