Isomorphism of fields

This article gives a basic definition in the following area: field theory
View other basic definitions in field theory |View terms related to field theory |View facts related to field theory

Definition

Definition from first principles

Suppose $K$ and $L$ are fields. A function $\varphi :K\to L$ is termed an isomorphism of fields if $\varphi$ is bijective (i.e., it is both injective and surjective) and satisfies the following three conditions:

1. $\varphi (1)=1$
2. $\varphi (a+b)=\varphi (a)+\varphi (b)\ \forall a,b\in K$
3. $\varphi (ab)=\varphi (a)\varphi (b)\ \forall a,b\in K$

If an isomorphism exists between two fields, we say that the fields are isomorphic.

As a ring isomorphism

Two fields $K$ and $L$ are isomorphic as fields with field isomorphism $\varphi$ if $K$ and $L$ are isomorphic as rings with ring isomorphism $\varphi$.