When does a polynomial with integer coefficients have only irrational roots?

Question

Given a polynomial $P$ with integer coefficients, is there any simple criterion (other than explicitly calculating the roots) with which I can check whether its roots are irrational?

Answer 1

Before improvements you can obtain a bound in an elementary way

$$f(x) = \sum_{n=0}^N a_n x^n \in \mathbb{Z}[x]$$ Then $$g(x)= a_N^{N-1} f(x/a_N)= x^N+\sum_{n=0}^{N-1} b_nx^n \in \mathbb{Z}[x] $$

• $M = \sup_n |b_n|$. If $|x| > N M$ then $g(x) \ne 0$.

• If $g(t) = 0, t=p/q \in \mathbb{Q}$ then $t^N = -\sum_{n=0}^{N-1} b_n t^n $ $$\implies \mathbb{Z}[t] \subset \{ \sum_{n=0}^{N-1} c_n t^n, c_n \in \mathbb{Z}\} \subset q^{-N} \mathbb{Z}$$ This is a contradiction if $t \not \in \mathbb{Z}$. Thus $t \in \mathbb{Z}$ and $t \in \{-NM,\ldots,NM\}$.