Content of Monic Polynomial

Theorem
Let $f$ be a monic polynomial with rational coefficients.

Let $\operatorname{cont}\left({f}\right)$ be the content of $f$.

Then $\operatorname{cont}\left({f}\right) = 1/n$ for some integer $n$.

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

Otherwise, let


 * $ f = X^r + \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 $X^r$, say $n = kd$ with $k \in \Z$.

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