Division Theorem for Polynomial Forms over Field/Proof 3

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Theorem

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $X$ be transcendental over $F$.

Let $F \sqbrk X$ be the ring of polynomials in $X$ over $F$.

Let $d$ be an element of $F \sqbrk X$ of degree $n \ge 1$.


Then $\forall f \in F \sqbrk X: \exists q, r \in F \sqbrk X: f = q \circ d + r$ such that either:

$(1): \quad r = 0_F$

or:

$(2): \quad r \ne 0_F$ and $r$ has degree that is less than $n$.


Proof

Proof by induction:

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:

$\forall f \in F \sqbrk X: \exists q, r \in F \sqbrk X: f = q \circ d + r$ provided that $\map \deg f < n$


Basis for the Induction

$\map P 0$ is the statement that $q$ and $r$ exist when $f = 0$.

This is shown trivially to be true by taking $q = r = 0$.

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.


So this is our induction hypothesis:

$\forall f \in F \sqbrk X: \exists q, r \in F \sqbrk X: f = q \circ d + r$ provided that $\map \deg f < k$


Then we need to show:

$\forall f \in F \sqbrk X: \exists q, r \in F \sqbrk X: f = q \circ d + r$ provided that $\map \deg f < k + 1$


Induction Step

This is our induction step:

Let $f$ be such that $\map \deg f = n$.

Let:

$f = a_0 + a_1 \circ x + a_2 \circ x^2 + \cdots + a_n \circ x^n$ where $a_n \ne 0$

Let:

$d = b_0 + b_1 \circ x + b_2 \circ x^2 + \cdots + b_j \circ x^j$ where $b_j \ne 0$

If $n < l$ then take $q = 0, r = f$.

If $n \ge l$, consider:

$c := f - a_n b_j^{-1} x^{n - j} d$

This has been carefully arranged so that the coefficient of $x^n$ in $c$ is zero.

Thus $\map \deg c < n$.

Therefore, by the induction hypothesis:

$c = d q_0 + r$

where $\map \deg r < \map \deg d$.

Therefore:

\(\displaystyle f\) \(=\) \(\displaystyle d \paren {q_0 + a_n b_j^{-1} x^{n - j} } + r\)
\(\displaystyle \) \(=\) \(\displaystyle d q + r\) where $\map \deg r < \map \deg d$ and $q = q_0 + a_n b_j^{-1} x^{n - j}$

Thus the existence of $q$ and $r$ have been established.


As for uniqueness, assume:

$d q + r = d q' + r'$

with $\map \deg r < \map \deg d, \map \deg {r'} < \map \deg d$

Then:

$d \paren {q - q'} = r' - r$

By Degree of Sum of Polynomials:

$\map \deg {r' - r} \le \max \set {\map \deg {r'}, \map \deg r} < \map \deg d$

and by Degree of Product of Polynomials over Integral Domain:

$\map \deg {d \paren {q - q'} } = \map \deg d + \map \deg {q - q'}$

That is:

$\map \deg d < \map \deg d + \map \deg {q - q'}$

and the only way for that to happen is for:

$\map \deg {q - q'} = -\infty$

that is, for $q - q'$ to be the null polynomial.

That is, $q - q' = 0_F$ and by a similar argument $r' - r = 0_F$, demonstrating the uniqueness of $q$ and $r$.

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall f \in F \sqbrk X: \exists q, r \in F \sqbrk X: f = q \circ d + r$ such that either:
$(1): \quad r = 0_F$
or:
$(2): \quad r \ne 0_F$ and $r$ has degree that is less than $n$.

$\blacksquare$


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