Definition:Field Homomorphism

Definition

Let $\struct {F, +, \times}$ and $\struct {K, \oplus, \otimes}$ be fields.

Let $\phi: F \to K$ be a mapping such that both $+$ and $\times$ have the morphism property under $\phi$.

That is, $\forall a, b \in F$:

\(\text {(1)}: \quad\)

\(\ds \map \phi {a + b}\)

\(=\)

\(\ds \map \phi a \oplus \map \phi b\)

\(\text {(2)}: \quad\)

\(\ds \map \phi {a \times b}\)

\(=\)

\(\ds \map \phi a \otimes \map \phi b\)

Then $\phi: \struct {F, +, \times} \to \struct {K, \oplus, \otimes}$ is a field homomorphism.

Also see

• Definition:Homomorphism (Abstract Algebra)
• Definition:Ring Homomorphism

• Definition:Field Epimorphism: a surjective field homomorphism

• Definition:Field Monomorphism: an injective field homomorphism

• Definition:Field Isomorphism: a bijective field homomorphism

• Definition:Field Endomorphism: a field homomorphism from a field to itself

• Definition:Field Automorphism: a field isomorphism from a field to itself

• Results about field homomorphisms can be found here.

Linguistic Note

The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.

Sources

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