PlanetMath: field homomorphism

(more info)

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field homomorphism

(Definition)

Let and be fields.

Definition 1 A field homomorphism is a function $\psi\colon F \to K$ such that:

$\psi(a+b) = \psi(a)+\psi(b)$ for all $a,b \in F$
$\psi(a\cdot b) = \psi(a) \cdot \psi(b)$ for all $a,b \in F$
$\psi(1)=1,\quad \psi(0)=0$

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

Lemma 1 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

only has two ideals, namely $\{0\}, F$ . Moreover, by the definition of field homomorphism, $\psi(1)=1$ , hence

is not in the kernel of the map, so the kernel must be equal to $\{0\}$ . $\qedsymbol$

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

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

so there is a “copy” of

. In other words, if

$\displaystyle \psi\colon F\to K$

is a field homomorphism then there exist a subfield

such that $H\cong F$ . Conversely, suppose there exists $H\subset K$ with

isomorphic to

. Then there is an isomorphism

$\displaystyle \chi \colon F \to H$

and we also have the inclusion homomorphism

$\displaystyle \iota\colon H \hookrightarrow K$

Thus the composition

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

is a field homomorphism.

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

"field homomorphism" is owned by alozano.

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