Definition:Isomorphism (Abstract Algebra)/Field Isomorphism

Definition

Let $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ be fields.

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

Then $\phi$ is a field isomorphism if and only if $\phi$ is a bijection.

That is, $\phi$ is a field isomorphism if and only if $\phi$ is both a monomorphism and an epimorphism.

Linguistic Note

The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.