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# field homomorphism

Let $F$ and $K$ be fields.

###### Definition.

A field homomorphism is a function $\psi\colon F\to K$ 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. $\psi(1)=1,\quad\psi(0)=0$

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

###### Lemma.

Let $\psi\colon F\to K$ 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

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

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

$\psi\colon F\to K$ |

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

$\chi\colon F\to H$ |

and we also have the inclusion homomorphism

$\iota\colon H\hookrightarrow K$ |

Thus the composition

$\iota\circ\chi\colon F\to K$ |

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.

## Mathematics Subject Classification

12E99*no label found*

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