4
$\begingroup$

Given a monic polynomial $f \in \mathbb{Z}[X]$, I would like to consider the ideal $(f, f')_{\mathbb{Z}[X]} \cap \mathbb{Z}$ in $\mathbb{Z}$. In particular: is it true that this is generated by the discriminant $\Delta(f)$?

I know at least that $\Delta(f)$ is contained in this ideal.

  • 0
    @Patrick, you might also enjoy the follow-up, Norms in polynomial rings, Bulletin of the Australian Mathematical Society 41, Issue 03, June 1990, pp 381-386. I'm not sure whether that's freely available.2012-11-27

1 Answers 1

3

No.

Take $f(x)=x^2+1$ as an example. Then $f'(x)=2x$ and $2=2f(x)-xf'(x) \in (f,f')_{\Bbb Z[X]} \cap \Bbb Z$, but $\Delta(f)=-4$.