Polynomial has Integer Coefficients iff Content is Integer

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

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

Then $f$ has integer coefficients if and only if $\operatorname{cont}\left({f}\right)$ is an integer.

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

Conversely, suppose that:


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

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 $\operatorname{cont}(f)\notin \Z$.