Homomorphism of fields

This article gives a basic definition in the following area: field theory
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Definition

Definition from first principles

Suppose $K$ and $L$ are fields. A function $\phi :K\to L$ is termed a homomorphism of fields if $\phi$ satisfies the following three conditions:

1. $\phi (1)=1$
2. $\phi (a+b)=\phi (a)+\phi (b)\mathrm{\forall }a,b\in K$
3. $\phi (ab)=\phi (a)\phi (b)\mathrm{\forall }a,b\in K$

Related notions

• If a field homomorphism is bijective, it is called a field isomorphism.
• If a field homomorphism maps a field to itself, it is called a field endomorphism.
• If a field homomorphism is bijective and maps a field to itself, it is called a field automorphism.