Question

If a monic polynomial with integer coefficients has a root in $\mathrm{\mathbb{Q}}$ , show that this root must be an integer.

Short answer

It is proved that this root must be an integer.

Step-by-step solution

Rational Root test

Consider that $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$as a polynomial with integer coefficients. When$r\ne 0$, and the rational number $\frac{r}{s}$(In lowest terms) is a root of $f\left(x\right)$then$r/a_0$and $s/a_0$.

Show that this root must be an integer

Suppose that$f\left(x\right)$ would be a monic polynomial with integer coefficients and contain a root in$\mathrm{\mathbb{Q}}$ , denote it $\frac{r}{s}$for such integers $r$and $s$. Then $\frac{r}{a_0}$and $\frac{s}{a_0}$according to the Rational Root Test. Since $f\left(x\right)$would be monic and $\frac{s}{a_0}$, where $a_n=1$then, $s=1$. The root $\frac{r}{s}=\frac{r}{1}$would be an integer.

Hence, it is proved that this root must be an integer.