Suppose $f(x)\in\mathbb{Z}[x]$ be a monic quadratic polynomial given by $f(x)=x^2+bx+c$, where $b,c\in\mathbb{Z}$. It is easy to check that $f(x)$ has both the roots integers if and only if the discriminant $\Delta_{f(x)}=b^2-4c$ is a square of some integer.
What I am looking for is the following: Is there any (sufficient) condition on the coefficients of a general $n$ degree monic polynomial $f(x)=x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0$, where $a_i\in\mathbb{Z}, i=0,1,\ldots,n-1$, so that all the $n$ complex roots of $f(x)$ are integers?
In particular, can something be said when $\operatorname{Deg}(f(x))$ is some odd prime $p$?
Any help (especially if someone can provide me with some research articles regarding this problem) would be appreciated.
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SPDR
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Maybe this helps: the discriminant of a monic degree $n$ polynomial is $$\prod_{i<j}(r_i-r_j)^2$$ where $r_i$ are the roots. – cansomeonehelpmeout Jun 13 '25 at 12:59
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In particular, if you have a repeated root the discriminant is $0^2=0$ trivially – cansomeonehelpmeout Jun 13 '25 at 13:02
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1You might want to check this out as well – cansomeonehelpmeout Jun 13 '25 at 13:11
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@cansomeonehelpmeout What I understand is that if all the roots of a monic $n$ degree polynomial has integer roots then the discriminant must be a square. But that's a necessary condition for having integer roots. I was looking for a sufficient condition. – SPDR Jun 13 '25 at 17:47