974: Field Generated by Subset of Superfield over Subfield

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definition of field generated by subset of superfield over subfield

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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 field.

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Target Context

• The reader will have a definition of field generated by subset of superfield over subfield.

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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:
$F^{\prime }$: $\in \{\text{ the fields }\}$
$F$: $\in \{\text{ the fields }\}$ such that $F\subseteq F^{\prime }$
$S$: $\subseteq F^{\prime }$, just a subset
$\ast F_{F^{\prime }}(S)$: $=\text{ the smallest field }$ such that $F\cup S\subseteq F_{F^{\prime }}(S)\subseteq F^{\prime }$
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Conditions:
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2: Note

$F_{F^{\prime }}(S)$ is uniquely determined, because it is the intersection of the fields such that $F\cup S\subseteq F_{F^{\prime }}(S)\subseteq F^{\prime }$, while there is at least 1 such, $F^{\prime }$: refer to the proposition that for any ring and any set of subfields, the intersection of the set is a subfield.

$F_{F^{\prime }}(S)$ seems to be widely denoted as $F(S)$, but $F_{F^{\prime }}(S)$ is clearer because it depends on $F^{\prime }$.

$S\subseteq F$ is possible, and then, $F_{F^{\prime }}(S)=F$.

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References

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