# 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, $\forall a, b \in F$:

 $(1):\quad$ $\displaystyle \map \phi {a + b}$ $=$ $\displaystyle \map \phi a \oplus \map \phi b$ $(2):\quad$ $\displaystyle \map \phi {a \times b}$ $=$ $\displaystyle \map \phi a \otimes \map \phi b$

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

## Also see

• 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.