description/proof of that for field and subfield, polynomial and nonzero polynomial divisor over field with subfield coefficients have quotient and remainder with subfield coefficients
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Note 1
- 3: Proof
- 4: Note 2
Starting Context
- The reader knows a definition of field.
- The reader knows a definition of polynomials ring over commutative ring.
- The reader admits the proposition that over any field, any polynomial and any nonzero polynomial divisor have the unique quotient and remainder.
Target Context
- The reader will have a description and a proof of the proposition that for any field and any subfield, any polynomial and any nonzero polynomial divisor over the field both with only subfield coefficients have the quotient and the remainder both with only subfield coefficients.
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:
\(F'\): \(\in \{\text{ the fields }\}\)
\(F\): \(\in \{\text{ the subfields of } F\}\)
\(F' [x]\): \(= \text{ the polynomials ring over } F'\)
\(p' (x)\): \(\in F' [x]\)
\(p (x)\): \(\in F' [x]\), \(\neq 0\)
\(q (x)\): \(\in F' [x]\), \(= \text{ the quotient of } p' (x) / p (x)\)
\(r (x)\): \(\in F' [x]\), \(= \text{ the remainder of } p' (x) / p (x)\)
//
Statements:
\(\text{ the coefficients of } p' (x) \text{ and } p (x) \text{ are in } F\)
\(\implies\)
\(\text{ the coefficients of } q (x) \text{ and } r (x) \text{ are in } F\)
//
2: Note 1
As an immediate corollary, any polynomial and any nonzero polynomial divisor over the complex numbers field both with only real coefficients have the quotient and the remainder both with only real coefficients.
3: Proof
Whole Strategy: apply the proposition that over any field, any polynomial and any nonzero polynomial divisor have the unique quotient and remainder; Step 1: regard \(p' (x)\) and \(p (x)\) as in \(F [x]\), and have \(p' (x) = p (x) \widetilde{q} (x) + \widetilde{r} (x)\), where \(\widetilde{q} (x)\) and \(\widetilde{r} (x)\) are the quotient and the remainder; Step 2: see that \(\widetilde{q} (x)\) and \(\widetilde{r} (x)\) are also the quotient and the remainder with \(p' (x)\) and \(p (x)\) regarded as in \(F' [x]\), and conclude the proposition.
Step 1:
\(p' (x)\) and \(p (x)\) can be regarded to be in \(F [x]\), because the coefficients of them are in \(F\).
\(p' (x) = p (x) \widetilde{q} (x) + \widetilde{r} (x)\), where \(\widetilde{q} (x)\) and \(\widetilde{r} (x)\) are the quotient and the remainder in \(F [x]\), because we are talking in \(F [x]\), by the proposition that over any field, any polynomial and any nonzero polynomial divisor have the unique quotient and remainder.
Step 2:
But \(\widetilde{q} (x)\) and \(\widetilde{r} (x)\) can be regarded to be in \(F' [x]\), because they have \(F'\) coefficients.
Then, \(p' (x) = p (x) \widetilde{q} (x) + \widetilde{r} (x)\) implies that \(\widetilde{q} (x)\) and \(\widetilde{r} (x)\) are a quotient and a remainder with \(p' (x)\) and \(p (x)\) regarded as in \(F' [x]\), because \(deg (\widetilde{r} (x)) \lt deg (p (x))\) holds anyway.
But by the proposition that over any field, any polynomial and any nonzero polynomial divisor have the unique quotient and remainder, \(\widetilde{q} (x)\) and \(\widetilde{r} (x)\) are nothing but the unique quotient and the unique remainder.
As \(\widetilde{q} (x)\) and \(\widetilde{r} (x)\) have only \(F\) coefficients, the quotient and the remainder of \(p' (x) / p (x)\) have only \(F\) coefficients.
4: Note 2
Another proof, which is more direct, is that the proposition that over any field, any polynomial and any nonzero polynomial divisor have the unique quotient and remainder has shown that the coefficients of the quotient and the remainder are determined by the field operations from the coefficients of \(p' (x)\) and \(p (x)\), so, as the coefficients of \(p' (x)\) and \(p (x)\) are in \(F\), the results of the field operations are in \(F\), because \(F\) is closed under the field operations.
