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I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd like to know whether or not this is true, and if so, a hint as to how to prove it.

For $f,g\in\mathbb{Z}[X]$ splitting as $f=a\prod_{i=0}^m(X-\alpha_i)$ and $g=b\prod_{j=0}^n(X-\beta_j)$ over $\overline{\mathbb{Q}}$, I understand the resultant of $f$ and $g$ to be $$R(f,g):=a^nb^m\prod_{i=0}^m\prod_{j=0}^n(\alpha_i-\beta_j).$$

Edit: From the first few replies it is clear that I rushed this post. I should mention that for my own purposes I only require a proof for distinct $f$ and $g$, both monic and irreducible over $\mathbb{R}$. More explicitly; $f$ and $g$ are both either linear, or quadratic with negative discriminant.

It seemed to me that the desired result would generalise, though clearly some assumptions must be made. Perhaps it is enough to assume both $f$ and $g$ monic, and the $\alpha_i$ and $\beta_j$ all distinct?

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So we at least assume that $f,g$ have not common zero? For $f=g=x$... – wxu May 1 '12 at 16:45
Need an assumption on leading terms too, I think. Res(2x, 2x+1) = 2, but <2x, 2x+1> = <1>. – Hurkyl May 1 '12 at 16:54
As a partial answer, if $\mathcal{O} = \mathbb{Z}[X]/f$ is the ring of integers in a number field (and so $f$ is monic), then the norm of (the image of $g$ in $\mathcal{O}$) is given by $N(g) = R(f,g)$. However, the $|N(g)|$ is also the size of $\mathcal{O}/g$. – Hurkyl May 1 '12 at 16:55
I bet the answer is something like "If the curves defined by $f=0$ and $g=0$ in $\mathbb{P}^1_\mathbb{Z}$ intersect only at affine points, at non-singular points, and aren't tangent to each other at such points, then the conjecture holds". Might even be an if and only if! – Hurkyl May 1 '12 at 17:30
@Hurkyl, I published the equality of norm and resultant in On resultants, Proc Amer Math Soc 89 (1983) 419-420. I then discovered that the result had already been published half a dozen times, see my paper Norms in polynomial rings, Bull Austral Math Soc 41 (1990) 381-386. Anyway, I think the first paper would be of interest to OP. The second paper attempts a generalization to several variables. – Gerry Myerson May 2 '12 at 6:25

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