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 $ is injective.
Proof.
Indeed, if $\psi $ is a field homomorphism, in particular it is a ring homomorphism^{}. Note that the kernel of a ring homomorphism is an 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 be equal to $\{0\}$. ∎
Remark: For this reason the terms “field homomorphism” and “field monomorphism” are synonymous. Also note that if $\psi $ is a 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 a subfield^{} $H$ of $K$ such that $H\cong F$. Conversely, suppose there exists $H\subset K$ with $H$ isomorphic^{} to $F$. Then there is an isomorphism^{}
$$\chi :F\to H$$ 
and we also have the inclusion homomorphism^{}
$$\iota :H\hookrightarrow K$$ 
Thus the 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.
Title  field homomorphism 

Canonical name  FieldHomomorphism 
Date of creation  20130322 13:54:54 
Last modified on  20130322 13:54:54 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  9 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 12E99 
Synonym  field monomorphism 
Related topic  RingHomomorphism 
Defines  field homomorphism 
Defines  field isomorphism 