# Ideal generated by two coprime polynomials in $\mathbb{Z}[x]$

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:

• This was also posted to mathoverflow. Please note that crossposting between SE sites is highly frowned upon - try one site first, and if you don't get a satisfactory response, ask a moderator to migrate the question to a different site. If you insist on posting in many sites, at least provide links to the other posts - as you can imagine, it would be frustrating for someone to put time into answering your question here, only to find out that you'd already gotten an answer elsewhere. – Zev Chonoles Aug 29 '16 at 20:31
• With any luck, Gerry M. will answer on MO. He is in Australia and should be awake soon. You might as well just write to him. – Will Jagy Aug 29 '16 at 20:31
• @Zev Chonoles: ok thank you for the information! meanwhile, should i delete the post here? – KonKan Aug 29 '16 at 21:39
• @KonKan: You can, or you can also provide a link on each copy to the other copy. – Zev Chonoles Aug 29 '16 at 21:41
• you already did half the way. The other half is: mathoverflow.net/questions/248574/… – KonKan Aug 29 '16 at 21:53