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

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

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

## 1 Answer

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.

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