876: Integers Modulo Prime Number Field

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definition of integers modulo prime number 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 integers modulo natural number ring.
• The reader knows a definition of field.

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Target Context

• The reader will have a definition of integers modulo prime number 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:
$p$: $\in \{\text{ the prime numbers }\}$
$\ast \mathbb{Z}/p$: $=\text{ the integers modulo natural number ring }$, $\in \{\text{ the fields }\}$
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Conditions:
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Frequently, it is denoted also as ${\mathbb{F}}_p$: $\mathbb{Z}/p={\mathbb{F}}_p$.

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2: Note

$\mathbb{Z}/p$ is indeed a field, by the proposition that the quotient ring of the integers ring by any prime principal ideal is a field.

So, when $n$ is any prime number, the integers modulo natural number ring, $\mathbb{Z}/n$, inevitably becomes a field

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References

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