Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\mathbb{Z}$ which can be expressed as linear combinations of $f(x), g(x)$ with coefficients in $\mathbb{Z}[x]$. $I \cap \mathbb{Z}$ is clearly an ideal in $\mathbb{Z}$. The following argument shows that, **given two coprime integer polynomials $f,g$, the ideal $I \cap \mathbb{Z}$ of $\mathbb{Z}$ has always non zero elements**:


If $f,g$ are considered as elements of $\mathbb{Q}[x]$, Bezout's identity tells us that there exists a pair of unique rational polynomials $U(x)$, $V(x)$ with $\deg U<\deg g$, $\deg V<\deg f$ such that
$$
U(x)f(x)+V(x)g(x)=1
$$
Thus, clearing denominators in the above identity, we get that there exists a pair of unique integer polynomials $u(x)$, $v(x)$ with $\deg u<\deg g$, $\deg v<\deg f$ such that
$$
u(x)f(x)+v(x)g(x)=c
$$
where $c$ is the $lcm$ of the denominators of $U,V$ and: $u=cU$, $v=cV$. Thus: $0 \neq c\in I \cap \mathbb{Z}$.


**Question $1$:** Can we determine a sufficient and necessary condition such that: $\mathbb{Z}\subset I \cap \mathbb{Z}$? 


**Question $2$:** Is there a general method for determining the least positive generator of $I \cap \mathbb{Z}$ ? 


*References:* 


* [Reduced resultants and Bezout's identity](https://mathoverflow.net/questions/248488/reduced-resultants-and-bezouts-identity)
* [The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$](https://mathoverflow.net/questions/17501/the-resultant-and-the-ideal-generated-by-two-polynomials-in-mathbbzx?noredirect=1&lq=1)
* [Reduced resultant of monic polynomials](https://mathoverflow.net/questions/227227/reduced-resultant-of-monic-polynomials?noredirect=1&lq=1)


**P.S.:** The second question above, is actually part of the question posted in the first of the references above. However, it is posted here rather as a problem of commutative algebra than a problem of number theory. 



Ideal generated by two univariate, coprime, integer polynomials
For question 2, compute a Gröbner basis over $\mathbb{Z}$ for the ideal generated by $f(x)$ and $g(x)$, which gives the required generator. You can do this easily in SageMathCloud (available free to use). The algorithm behind this computation is exactly what you would be doing by hand to find the generator. This also answers question 1, if you allow an algorithm to provide a necessary and sufficient condition.


