I was asked the following question by a colleague and was embarrassed not to know the answer.


Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, the elements of $\mathbb{Z}$ which are linear combinations of $f(x), g(x)$ with coefficients in $\mathbb{Z}[x]$. Then $I$ is clearly an ideal in $\mathbb{Z}$. Let $D>0$ be a generator of this ideal. The question is: what is $D$?


Now, we do have the standard resultant $R$ of $f,g$, which under our hypotheses, is a non-zero integer. We know that $R \in I$ and it's not hard to show that a prime divides $R$ if and only if it divides $D$. I thought $R = \pm D$ but examples show that this is not the case. 



The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$
This quantity $D$ is known as the "congruence number" or "reduced resultant" of the polynomials f and g. I first saw this in a preprint by Wiese and Taixes i Ventosa, <http://arxiv.org/abs/0909.2724>. They ascribe the concept to a paper which I don't have a copy of: 


M. Pohst. *A note on index divisors*. In *Computational number theory (Debrecen, 1989)*, 173–
182, de Gruyter, Berlin, 1991.


