Factoring polynomials

Definition 5.1   If $f\in \mathbb{Z}[x]$, we let the norm of $f$ be the norm of the vector of its coefficients. That is,

$$\Vert x\mapsto ax^2+bx+c\Vert=\sqrt{a^2+b^2+c^2}$$

Exercise 5.2 (Mignotte's Lemma)   If $g, f\in \mathbb{Z}[x]$, and $g{\,\mid\,}f$, then $\Vert g\Vert\leq 2^{\text{deg}(g)}\Vert f\Vert$.