Gauss's Lemma on Primitive Rational Polynomials/Proof 2

Theorem
Let $\Q \left[{X}\right]$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$.

Let $f \left({X}\right), g \left({X}\right) \in \Q \left[{X}\right]$ be primitive polynomials.

Then the product of $f$ and $g$ is also a primitive polynomial.

Proof
Recall Polynomial has Integer Coefficients iff Content is Integer.

By hypothesis $f$ and $g$ have content $1 \in \Z$, so $f, g \in \Z \left[{X}\right]$.

Suppose that $f g$ is not primitive, say:


 * $\operatorname{cont} \left({f g}\right) = d \neq 1$

By the Fundamental Theorem of Arithmetic we can choose a prime $p$ dividing $d$.

Let $\pi : \Z \to \Z / p \Z$ be the canonical epimorphism to the ring of integers modulo $p$.

From Ring of Integers Modulo Prime is Field, $\Z / p \Z$ is a field.

Let $\Pi : \Z \left[{X}\right] \to \left({\Z / p \Z}\right) \left[{X}\right]$ be the induced homomorphism of the polynomial rings.

By construction, $p$ divides each coefficient of $f g$, so:


 * $\Pi \left({f g}\right) = \Pi\left({f}\right) \Pi\left({g}\right) = 0$.

Since a polynomial ring over a field is an integral domain we know that $\left({\Z / p \Z}\right) \left[{X}\right]$ is an integral domain.

Thus we must have $\Pi \left({f}\right) = 0$ or $\Pi \left({g}\right) = 0$.

After possibly exchanging $f$ and $g$, we may assume that $\Pi \left({f}\right) = 0$.

Now by the characterization of the kernel of the induced homomorphism, if:


 * $f = a_0 + a_1 X + \cdots + a_n X^n$

we must have:


 * $\pi \left({a_i}\right) = 0$ for $i = 0, \ldots, n$.

That is, $a_i \in p \Z$ for $i = 0, \ldots, n$.

But this says precisely that $p$ divides each $a_i$, $i = 0, \ldots, n$.

Therefore $p$ divides the content of $f$, a contradiction.