980: Polynomial Extended over Extended Field

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definition of polynomial extended over extended field

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Topics

About: ring

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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 polynomials ring over commutative ring.
• The reader knows a definition of extended field of field.

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Target Context

• The reader will have a definition of polynomial extended over extended 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:
$F$: $\in \{\text{ the fields }\}$
$F^{\prime }$: $\in \{\text{ the extended fields of }F\}$
$F[x]$: $=\text{ the polynomials ring over }F$
$F^{\prime }[x]$: $=\text{ the polynomials ring over }{F}^{\prime }$
$p(x)$: $\in F[x]$
$\ast \overline{p(x)}$: $\in F^{\prime }[x]$, $=p(x)\text{ regarded to be in }{F}^{\prime }[x]$
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Conditions:
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2: Note

The coefficients of $p(x)$ are in $F$ and are in $F^{\prime }$, and so, $p(x)$ can be regarded to be in $F^{\prime }[x]$, which this definition is saying.

$\overline{p(x)}$ is sometimes (or rather usually) denoted as $p(x)$, but sometimes we need to distinguish $\overline{p(x)}$ from $p(x)$: for example, it may be that $p(x)$ is irreducible while $\overline{p(x)}$ is not irreducible.

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References

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