960: Bijective Field Homomorphism Is 'Fields - Homomorphisms' Isomorphism

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description/proof of that bijective field homomorphism is 'fields - homomorphisms' isomorphism

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Topics

About: field

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note
• 3: Proof

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Starting Context

• The reader knows a definition of field.
• The reader knows a definition of bijection.
• The reader knows a definition of %structure kind name% homomorphism.
• The reader knows a definition of %category name% isomorphism.
• The reader admits the proposition that any bijective ring homomorphism is a 'rings - homomorphisms' isomorphism.

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Target Context

• The reader will have a description and a proof of the proposition that any bijective field homomorphism is a 'fields - homomorphisms' isomorphism.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F_1$: $\in \{\text{ the fields }\}$
$F_2$: $\in \{\text{ the fields }\}$
$f$: $F_1\to F_2$, $\in \{\text{ the bijections }\}\cap \{\text{ the field homomorphisms }\}$
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Statements:
$f\in \{\text{ the 'fields - homomorphisms' isomorphisms }\}$
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2: Note

Sometimes, the definition that dictates any bijective field homomorphism to be a 'fields - homomorphisms' isomorphism is seen, but that does not seem a good practice: 'isomorphism' is a general concept defined in the category theory and requires the inverse to be a homomorphism in the category.

In general, a bijective morphism of a category is not necessarily any %category name% isomorphism. For example, a bijective continuous map, which is a morphism of the 'topological spaces - continuous maps' category, is not necessarily a homeomorphism, which is a 'topological spaces - continuous maps' isomorphism.

As this proposition holds, some people think that that definition that requires only bijective-ness is valid, but just because this proposition holds does not mean that the general definition made in the category theory should be deformed for the field case.

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3: Proof

Whole Strategy: Step 1: take the inverse, $f^{-1}$; Step 2: see that $f^{-1}$ is a field homomorphism.

Step 1:

As $f$ is bijective, there is the inverse, $f^{-1}:F_2\to F_1$.

Step 2:

Let us see that $f^{-1}$ is necessarily field homomorphic.

$F_1$ and $F_2$ are some rings and $f$ is a bijective ring homomorphism between the rings.

By the proposition that any bijective ring homomorphism is a 'rings - homomorphisms' isomorphism, $f$ is a 'rings - homomorphisms' isomorphism between the rings. So, $f^{-1}$ is a ring homomorphism between the rings.

The remaining issue is only whether for each element of $F_2$, $f^{-1}$ maps its inverse to the inverse of its image under $f^{-1}$.

Let $f(r_1)\in F_2$ be any. $f(r_1{)}^{-1}=f(r_1^{-1})$, because $f$ is field homomorphic. $f^{-1}(f(r_1{)}^{-1})=f^{-1}(f(r_1^{-1}))=r_1^{-1}={f^{-1}(f(r_1))}^{-1}$.

So, $f^{-1}$ is a field homomorphism.

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References

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