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[parent] field homomorphism (Definition)

Let $F$ and $K$ be fields.

Definition 1   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 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 $F$ 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 $\{0\}$ $ \qedsymbol$

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.




"field homomorphism" is owned by alozano.
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See Also: ring homomorphism

Other names:  field monomorphism
Also defines:  field homomorphism, field isomorphism
Keywords:  field, map

This object's parent.

Attachments:
field homomorphisms fix prime subfields (Theorem) by Wkbj79
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Cross-references: natural numbers, characteristic, composition, homomorphism, inclusion, isomorphism, isomorphic, conversely, subfield, terms, map, ideal, kernel, ring homomorphism, surjective, injective, function, fields
There are 12 references to this entry.

This is version 6 of field homomorphism, born on 2003-08-29, modified 2007-01-04.
Object id is 4670, canonical name is FieldHomomorphism.
Accessed 13538 times total.

Classification:
AMS MSC12E99 (Field theory and polynomials :: General field theory :: Miscellaneous)

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