field homomorphism

Let $F$ and $K$ be fields.

Definition.

A field homomorphism is a function $\psi \mathrm{:}F\mathrm{\to }K$ such that:

1. $\psi (a+b)=\psi (a)+\psi (b)$ for all $a,b\in F$

2. $\psi (a\cdot b)=\psi (a)\cdot \psi (b)$ for all $a,b\in F$

3. $\psi (1)=1,\psi (0)=0$

If $\psi$ is injective and surjective, then we say that $\psi$ is a field isomorphism.

Lemma.

Let $\psi \mathrm{:}F\mathrm{\to }K$ be a field homomorphism. Then $\psi$ isinjective.

Proof.

Indeed, if $\psi$ is a field homomorphism, in particular it is aring homomorphism. Note that the kernel of a ring homomorphism isan ideal and a field $F$ only has two ideals, namely $\{0\},F$.Moreover, by the definition of field homomorphism, $\psi (1)=1$,hence $1$ is not in the kernel of the map, so the kernel must beequal to $\{0\}$.∎

Remark: For this reason the terms “field homomorphism” and“field monomorphism” are synonymous. Also note that if $\psi$ isa field monomorphism, then

$$\psi (F)\cong F,\psi (F)\subseteq K$$

so there is a “copy” of $F$ in $K$. In other words, if

$$\psi :F\to K$$

is a field homomorphism then there exist asubfield $H$ of $K$ such that $H\cong F$. Conversely, supposethere exists $H\subset K$ with $H$ isomorphic to $F$. Then thereis an isomorphism

$$\chi :F\to H$$

and we also have theinclusion homomorphism

$$\iota :H\hookrightarrow K$$

Thusthe composition

$$\iota \circ \chi :F\to K$$

is a field homomorphism.

Remark: Let $\psi :F\to K$ be a field homomorphism. We claim that the characteristic of $F$ and $K$ must be the same. Indeed, since $\psi (1_F)=1_K$ and $\psi (0_F)=0_K$ then $\psi (n\cdot 1_F)=n\cdot 1_K$ for all natural numbers $n$. If the characteristic of $F$ is $p>0$ then $0=\psi (p\cdot 1)=p\cdot 1$ in $K$, and so the characteristic of $K$ is also $p$. If the characteristic of $F$ is $0$, then the characteristic of $K$ must be $0$ as well. For if $p\cdot 1=0$ in $K$ then $\psi (p\cdot 1)=0$, and since $\psi$ is injective by the lemma, we would have $p\cdot 1=0$ in $F$ as well.