Properties of Content

Theorem
The content of a polynomial satisfied the following:


 * 1) $\operatorname{cont}(f) \in \Z \iff f \in \Z[X]$
 * 2) If $f$ is monic then $\operatorname{cont}(f) = n^{-1}$ for some $n \in \N$
 * 3) Gauss's lemma: $\operatorname{cont}(fg) = \operatorname{cont}(f)\operatorname{cont}(g)$ for all $f,g\in \Q[X]$

Proof
1. Clearly if $f \in \Z[X]$ then $\operatorname{cont}(f) \in \Z$.

Conversely, suppose that $\operatorname{cont}(f) \in \Z$, with:


 * $ f = a_d X^d + \cdots + a_1X + a_0 \notin \Z[X]$

Let $n = \inf\left\{ n \in \N : nf \in \Z[X] \right\}$.

We must have that the greatest common divisor of $na_d,\ldots,na_0 < n$, otherwise we could take a smaller $n$.

Therefore $0 < \operatorname{cont}(f) < 1$, and $\operatorname{cont}(f)\notin \Z$.

2. If $f \in \Z[X]$ the result is trivial.

Otherwise, let


 * $ f = X^d + \cdots + a_1X + a_0 \notin \Z[X]$

and $n = \inf\left\{ n \in \N : nf \in \Z[X] \right\}$.

Then $d = \operatorname{cont}(nf) $ divides $n$, because $n$ is the coefficient of $X^d$, say $n = kd$.

Therefore $n^{-1}\operatorname{cont}(f) d n^{-1} = k^{-1}$

3. This is immediate from Gauss's lemma.