definition of field generated by subset of superfield over subfield
Topics
About: field
The table of contents of this article
Starting Context
- The reader knows a definition of field.
Target Context
- The reader will have a definition of field generated by subset of superfield over subfield.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F'\): \(\in \{\text{ the fields }\}\)
\( F\): \(\in \{\text{ the fields }\}\) such that \(F \subseteq F'\)
\( S\): \(\subseteq F'\), just a subset
\(*F_{F'} (S)\): \(= \text{ the smallest field }\) such that \(F \cup S \subseteq F_{F'} (S) \subseteq F'\)
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Conditions:
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2: Note
\(F_{F'} (S)\) is uniquely determined, because it is the intersection of the fields such that \(F \cup S \subseteq F_{F'} (S) \subseteq F'\), while there is at least 1 such, \(F'\): refer to the proposition that for any ring and any set of subfields, the intersection of the set is a subfield.
\(F_{F'} (S)\) seems to be widely denoted as \(F (S)\), but \(F_{F'} (S)\) is clearer because it depends on \(F'\).
\(S \subseteq F\) is possible, and then, \(F_{F'} (S) = F\).
