Consider the polynomial $$p(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ where all the coefficients $a_i$ are integers.

If $p(x)$ has a rational root $u/v$ where $\gcd(u,v)=1$ , then $u| a_0$ and $v| a_n$ .

This theorem is related to the result about monic polynomials whose coefficients belong to a unique factorization domain. Such theorem then states that any root in the fraction field is also in the base domain.