Definition:Field Homomorphism

Definition

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

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

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

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \phi \left({a + b}\right)\)	\(=\)	\(\displaystyle \phi \left({a}\right) \oplus \phi \left({b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \phi \left({a \times b}\right)\)	\(=\)	\(\displaystyle \phi \left({a}\right) \otimes \phi \left({b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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

Also see

• Homomorphism
• Ring Homomorphism

• Field Epimorphism: a surjective field homomorphism

• Field Monomorphism: an injective field homomorphism

• Field Isomorphism: a bijective field homomorphism

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

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

• Results about field homomorphisms can be found here.