Past exam of the algebra 2 course of the University of Oxford / 2014 / 2 / b / Solution

Write a rational root of a polynomial as $\gamma =a/b$ in lowest terms with $b>0$. Substituting into the monic integer-coefficient polynomial and multiplying by $b^d$ givesEvery term other than $a^d$ is divisible by $b$, so $b\mid a^d$. But $\gcd (a,b)=1$ implies $\gcd (a^d,b)=1$, and hence $b=1$. ThusThis is the monic case of the rational root theorem. The monic polynomial assumption is essential because it removes any leading coefficient that could supply prime factors of the denominator.