# Rational Polynomial is Content Times Primitive Polynomial/Existence

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## Theorem

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

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

Then:

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

where:

- $\cont f$ is the content of $\map f X$
- $\map {f^*} X$ is a primitive polynomial.

## Proof

Consider the coefficients of $f$ expressed as fractions.

Let $k$ be any positive integer that is divisible by the denominators of all the coefficients of $f$.

Such a number is bound to exist: just multiply all those denominators together, for example.

Then $\map f X$ is a polynomial equal to $\dfrac 1 k$ multiplied by a polynomial with integral coefficients.

Let $d$ be the GCD of all these integral coefficients.

Then $\map f X$ is equal to $\dfrac h k$ multiplied by a primitive polynomial.

$\blacksquare$

## 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 $61$