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