Gauss's Lemma on Primitive Rational Polynomials/Proof 2

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Theorem

Let $\Q$ be the field of rational numbers.

Let $\Q \sqbrk X$ be the ring of polynomials over $\Q$ in one indeterminate $X$.

Let $\map f X, \map g X \in \Q \sqbrk X$ be primitive polynomials.


Then their product $f g$ is also a primitive polynomial.


Proof

Recall Polynomial has Integer Coefficients iff Content is Integer.

By hypothesis $f$ and $g$ have content $1 \in \Z$.

Therefore:

$f, g \in \Z \sqbrk X$

Aiming for a contradiction, suppose that $f g$ is not primitive, say:

$\cont {f g} = d \ne 1$

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 $\Pi : \Z \sqbrk X \to \paren {\Z / p \Z} \sqbrk X$ be the induced homomorphism of the polynomial rings.

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

$\map \Pi {f g} = \map \Pi f \, \map \Pi g = 0$

From Polynomial Forms over Field form Integral Domain, $\paren {\Z / p \Z} \sqbrk X$ is an integral domain.

Thus:

$\map \Pi f = 0$ or $\map \Pi g = 0$

After possibly exchanging $f$ and $g$, we may assume that $\map \Pi f = 0$.


Now by Kernel of Induced Homomorphism of Polynomial Forms, if:

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

we must have:

$\map \pi {a_i} = 0$

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.

Hence our assumption that $f g$ is not primitive was invalid.

The result follows by Proof by Contradiction.

$\blacksquare$