Field Homomorphism is either Trivial or Injection/Proof 2

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

Suppose $\phi$ is not an injection.

So there must exist $a, b \in F: \map \phi a = \map \phi b$.

Let $k = a +_E \paren {-b}$.

Then:

As $a \ne b$ then $k \ne 0_E$ and so has a product inverse $\exists k^{-1} \in E$.

So for any $x \in E$ we can write $x = k \circ \paren {k^{-1} \circ x}$ and so:

So if $\phi$ is not an injection, it is the trivial homomorphism.