870: Field

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definition of field

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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

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Starting Context

• The reader knows a definition of ring.

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Target Context

• The reader will have a definition of field

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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:
$\ast F$: $\in \{\text{ the rings }\}$
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Conditions:
$\mathrm{\forall }{r}_1,r_2\in F(r_1{r}_2=r_2{r}_1)$
$\wedge$
$\mathrm{\forall }r\in F(\mathrm{\exists }{r}^{\prime }\in F(r{r}^{\prime }=r^{\prime }r=1))$
//

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2: Note

Inevitably, such $r^{\prime }$ is unique for fixed $r$, because supposing there is another $r^{″}\in F$ such that $r{r}^{″}=r^{″}r=1$, from $r^{\prime }r=1$, $r^{\prime }r{r}^{″}=1r^{″}=r^{″}$, but $r^{\prime }r{r}^{″}=r^{\prime }1=r^{\prime }$, and so, $r^{\prime }=r^{″}$.

As $r^{\prime }$ is unique for $r$, it is warranted to be denoted as $r^{-1}$, called "the inverse of $r$".

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References

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