686: Field Is Integral Domain

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description/proof of that field is integral domain

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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: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of field.
• The reader knows a definition of integral domain.

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Target Context

• The reader will have a description and a proof of the proposition that any field is an integral domain.

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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:
$F$: $\in \{\text{ the fields }\}$
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Statements:
$F\in \{\text{ the integral domains }\}$
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2: Natural Language Description

For any field, $F$, $F$ is an integral domain.

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3: Proof

Whole Strategy: Step 1: see that $F$ is a nonzero commutative ring; Step 2: see that for any 2 elements of $F$ such that the product is $0$, one of the elements is $0$.

Step 1:

$F$ is a nonzero commutative ring, by the definition of field.

Step 2:

Let us take any elements, $r_1,r_2\in F$. Let us suppose that $r_1{r}_2=0$. If $r_2\ne 0$, $r_1=r_1{r}_2{r_2}^{-1}=0{r_2}^{-1}=0$; if $r_1\ne 0$, $r_2={r_1}^{-1}{r}_1{r}_2={r_1}^{-1}0=0$. That means that $r_1=0$ or $r_2=0$.

As the logical contraposition, $\mathrm{\neg }(r_1=0\vee r_2=0)=(r_1\ne 0\wedge r_2\ne 0)⟹r_1{r}_2\ne 0$.

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References

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