Let $E$ and $F$ be fields.
Let $\phi: E \to F$ be a (field) homomorphism.
Let $\phi$ be a surjection.
Then $\phi$ is an isomorphism.
As asserted, let $\phi$ be a surjection.
From Field Homomorphism is either Trivial or Injection, $\phi$ is either an injection or the trivial homomorphism.
If $\phi$ is an injection, then, by definition, $\phi$ is a bijection.
Hence, again by definition, $\phi$ is an isomorphism.
If $\phi$ is not an injection, then $\phi$ is the trivial homomorphism.
But from Field Contains at least 2 Elements, $\Img \phi$ cannot in that case be a field.
Hence if $\phi$ is not an injection, then $\phi$ cannot be a surjection.
Hence the result.