# Definition:Isomorphism (Abstract Algebra)/Field Isomorphism

## Definition

Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ 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.