Content of Monic Polynomial

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Theorem

Let $f$ be a polynomial with rational coefficients.

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

If $f$ is monic, then $\operatorname{cont}\left({f}\right) = \dfrac 1 n$ for some integer $n$.


Proof

Since $f$ is monic, it can be written as:

$ f = X^r + \cdots + a_1 X + a_0$

Now let $n = \inf \left\{{ n \in \N : n f \in \Z \left[{X}\right] }\right\}$.

Let $d = \operatorname{cont} \left({n f}\right)$.

Then by definition of content:

$d = \gcd \left\{{n, n a_{r-1}, \ldots, n a_1, n a_0}\right\}$

Therefore, by definition of GCD, $d$ divides $n$.

So say $n = k d$ with $k \in \Z$.

Then:

$\operatorname{cont} \left({f}\right) = \dfrac d {k d} = \dfrac 1 k$

as desired.

$\blacksquare$