definition of polynomial extended over extended field
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of polynomials ring over commutative ring.
- The reader knows a definition of extended field of field.
Target Context
- The reader will have a definition of polynomial extended over extended field.
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( F'\): \(\in \{\text{ the extended fields of } F\}\)
\( F [x]\): \(= \text{ the polynomials ring over } F\)
\( F' [x]\): \(= \text{ the polynomials ring over } F'\)
\( p (x)\): \(\in F [x]\)
\(*\overline{p (x)}\): \(\in F' [x]\), \(= p (x) \text{ regarded to be in } F' [x]\)
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Conditions:
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2: Note
The coefficients of \(p (x)\) are in \(F\) and are in \(F'\), and so, \(p (x)\) can be regarded to be in \(F' [x]\), 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.
