Endomorphism of a field

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article gives a basic definition in the following area: field theory
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This article gives a basic definition in the following area: Galois theory
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This article defines a function property, viz a property of functions from a group to itself

Definition

Symbol-free definition

An endomorphism of a field is any of the following equivalent things:

• An homomorphism from the field to itself

Definition with symbols

Let $K$ be a field. A map $\sigma$ from $K$ to itself is termed an endomorphism of $K$ if it satisfies all of the following conditions:

• $\sigma (ab)=\sigma (a)\sigma (b)$ whenever $a$ and $b$ are both in $K$
• $\sigma (a+b)=\sigma (a)+\sigma (b)$ whenever $a$ and $b$ are both in $K$
• $\sigma (1)=1$

From these we can further deduce:

• $\sigma (0)=0$
• $\sigma (a^{-1})=\sigma (a)^{-1}$ whenever $a$ is in $K$
• $\sigma (-a)=\sigma (-a)$ whenever $a$ is in $K$.

See also

• Automorphism of a field
• Endomorphism of a group