Product of Rational Polynomials

Theorem

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

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

Using Rational Polynomial is Content Times Primitive Polynomial, let these be expressed as:

$f \left({X}\right) = c_f \cdot f^* \left({X}\right)$

$g \left({X}\right) = c_g \cdot g^* \left({X}\right)$

where:

• $c_f, c_g$ are the content of $f$ and $g$ respecively
• $f^*, g^*$ are primitive.

Let $h \left({X}\right) = f \left({X}\right) g \left({X}\right)$ be the product of $f$ and $g$.

Then:

• $c_h = c_f c_g$
• $h^* \left({X}\right) = f^* \left({X}\right) g^* \left({X}\right)$

Proof

We have, by applications of Rational Polynomial is Content Times Primitive Polynomial:

• $c_h \cdot h^* \left({X}\right) = c_f c_g \cdot f^* \left({X}\right) g^* \left({X}\right)$

By Product of Primitive Polynomials we have that $f^* \left({X}\right) g^* \left({X}\right)$ is primitive.

As $c_f > 0$ and $c_g > 0$, then so is $c_f c_g > 0$.

By the uniqueness clause in Rational Polynomial is Content Times Primitive Polynomial, the result follows.

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 6.31$: Theorem $62$