Surjective Field Homomorphism is Field Isomorphism

Theorem

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.

Proof

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.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87 \eta$