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.
$\blacksquare$