Rational Polynomial is Content Times Primitive Polynomial/Uniqueness

Theorem

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$.

Let $\map f X \in \Q \sqbrk X$ be given.

Then there exist unique content $\cont f$ of $\map f X$ and primitive polynomial $\map {f^*} X$ such that:

$\map f X = \cont f \, \map {f^*} X$

Proof

Existence is proved in Rational Polynomial is Content Times Primitive Polynomial: Existence.

Suppose that $a \cdot \map f X = b \cdot \map g X$ where $a, b \in \Q$ and $f, g$ are primitive.

Then:

$\map g X = \dfrac a b \map f X$

where $\dfrac a b$ is some rational number which can be expressed as $\dfrac m n$ where $m$ and $n$ are coprime.

Then:

$\map g X = \dfrac m n \map f X$

that is:

$m \cdot \map f X = n \cdot \map g X$

Suppose $m > 1$.

Then from Euclid's Lemma $m$ has a divisor $p$ which does not divide $n$ (as $m \perp n$).

So $m$ must divide every coefficient of $g$.

But this can not be so, as $g$ is primitive, so $m = 1$.

In a similar way, $n = 1$.

So $f = g$ and $a = b$, so demonstrating uniqueness.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 31$. Polynomials with Integer Coefficients: Theorem $61$