What are all necessary conditions on the roots for $P\in\mathbb{Z}[X]$?

Let $P(X)=a_0+a_1X+a_2X^2...+X^n$ and $S=\{s,P(s)=0\}\qquad n\geq2$

I found these conditions, are they right ? Is there anymore conditions?

If $s\in \mathbb{Z}\implies s |a_0 $

If $s\in \mathbb{C}\implies \overline{s}\in S$

If $P(X)>0\implies S\subset\mathbb{C} \quad n=2k, \; (k\in\mathbb{N}^*)$

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    $\begingroup$ The first one should be $s \in \mathbb{Z} \implies s \mid \frac{a_0}{a_n}$, no? $\endgroup$ – platty Aug 2 '17 at 15:45
  • $\begingroup$ Yes , I will correct this mistake $\endgroup$ – Stu Aug 2 '17 at 15:46
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    $\begingroup$ For more condition you can add the condition if the root is a rational $s = \frac{a}{b}$ then $a | a_0$ and $b | a_n$ $\endgroup$ – Youem Aug 2 '17 at 15:46
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    $\begingroup$ @ S. Ong I've got $a_n=1$ in my mind $\endgroup$ – Stu Aug 2 '17 at 15:50
  • $\begingroup$ @S. Ong , Ok, I understood, that is the Rational root theorem. Thanks $\endgroup$ – Stu Aug 2 '17 at 16:22

The roots of a polynomial don't determine its coefficients. For example, $X+1$ and $0.5X+0.5$ have the same roots.

Nevertheless, the set of the numbers that are roots of some polynomial in $\Bbb Z[X]$ is important, and it has a name: the set of algebraic numbers. So we could say that a necessary condition on the roots of $P$ for $P\in\Bbb Z[X]$ is that the roots are algebraic numbers. But this gives no criterion or information. It's only a name. There are wide areas of algebra dedicated to the algebraic numbers, though.

  • $\begingroup$ You mean the roots and $a_n$ determine the coefficients of polynomials, don't you? $\endgroup$ – Stu Aug 2 '17 at 16:02
  • $\begingroup$ @Stu I didn't exactly mean that, but it is true. $\endgroup$ – ajotatxe Aug 2 '17 at 16:03

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