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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.

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    How do you define $\Delta(f)$? There are many known definitions of the discriminant out there.2012-11-26
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    In On resultants, Proc. Amer. Math. Soc. 89 (1983), 419-420, available at http://www.ams.org/journals/proc/1983-089-03/S0002-9939-1983-0715856-2/ I proved that if $c$ generates your ideal then the resultant of $f$ and $f'$ divides $c^n$, where $n$ is the degree of $f$. As the discriminant can also be defined in terms of this resultant, you might find the paper useful.2012-11-26
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    @PatrickDaSilva I define it as the product over differences of zeros in $\mathbb{C}[X]$.2012-11-26
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    @GerryMyerson Thanks for the information!2012-11-26
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    @Gerry : I was reading your paper. You never define the "right regular representation of $\alpha$" in there. What is that?2012-11-26
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    As a further question: does the generator of this ideal have a special name or any significance in algebra?2012-11-26
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    @Patrick, given $a$ in $B$, the map $T_a:B\to B$ given by $T_a(b)=ab$ is linear. You can get a matrix for it by choosing a basis. The obvious choice is $1,x,x^2,\dots,x^{n-1}$, where $n$ is the degree of $f$; the matrix you get is called the right regular representation of $a$. A different choice of basis may yield a different matrix, but the determinant of the matrix is independent of the choice of basis.2012-11-26
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    The generator of the ideal has been called the *congruence number*, also the *reduced resultant*. See http://mathoverflow.net/questions/17501/the-resultant-and-the-ideal-generated-by-two-polynomials-in-mathbbzx for more information.2012-11-26
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    @GerryMyerson : Oh, that. Sure! Okay, makes more sense now. That paper of yours is very interesting! I wish I could have +1'd it, if that makes any sense. =P2012-11-26
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    @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

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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 $\Delta(f)=-4$.