2
$\begingroup$

Let $f,g \in \mathbb{Z}[x]$ both monic. Suppose that $\operatorname{Res}(f,g ) \neq 0 $, where $\operatorname{Res}(f,g) $ is the resultant of $f$ and $g$.

Is it true that $ \operatorname{discr}(g) | \operatorname{discr}(fg) $ ?

$\endgroup$
0

1 Answer 1

2
$\begingroup$

The answer is yes. Let $a_1,\ldots,a_m$ be the roots of $g$ and $a_{m+1},\ldots,a_n$ the roots of $f$ in some field extension $K$ of $\mathbb{Q}$. It follows that

$$\Delta(g)=\pm \prod_{i<j\leq m}(a_i-a_j)^2,\;\;\Delta(fg)=\pm \prod_{i<j\leq n}(a_i-a_j)^2=\Delta(g)\cdot\alpha,$$

where $\alpha$ has to be rational, since both discriminants are integers. However, $\alpha$ is also a polynomial in the roots $a_1,\ldots,a_n$, which are all integral over $\mathbb{Z}$ being roots of monic polynomials, so $\alpha$ is integral and rational. It follows that $\alpha$ must be in $\mathbb{Z}$ and the claim follows.

$\endgroup$
0

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .