Andre Weil noticed as a student in 1925 that the polynomial ring $\mathbb{Z}[x]$ comes close to being a PID, and he describes this as `` the embryo of my future thesis.''

He observed that, given $f(x),g(x)\in\mathbb{Z}[x]$, the Euclidean algorithm computes a sequence of polynomials where each is a linear combination of the preceding ones and which either:

1) Ends in a term which divides the preceding term and thus is something like a GCD for $f$ and $g$; or

2) Ends in an integer $d$ which does not divide the preceding term but bounds the common divisors of values pointwise in this way: For any integer $n$, an integer common divisor of $f(n),g(n)$ must divide $d$.

I have not seen this property discussed anywhere. Does it have a name?

There is related discussion in the question The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$

The number $d$ is called the reduced resultant of $f,g$.