Exercise 1026Number theory › IrrationalityL1Added 3 Nov 2026

Rational root theorem and irrationality

#rational root theorem#algebraic#irrationality

Statement

Let $P(X)=a_nX^n+\cdots+a_0\in\mathbb{Z}[X]$ with $a_n\neq0$ . Show that any rational root of $P$ is of the form $p/q$ with $p\mid a_0$ and $q\mid a_n$ .

Hint

Write $P(p/q)=0$ and multiply by $q^n$ ; reduce modulo $p$ and modulo $q$ .

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Solution

Setup. $P(p/q)=0$ , $\gcd(p,q)=1$ .

Multiply. $a_np^n+a_{n-1}p^{n-1}q+\cdots+a_0q^n=0$ .

$p\mid a_0$ . Mod $p$ : $a_0q^n\equiv0\pmod p$ . Since $\gcd(p,q)=1$ , $p\mid a_0$ .

$q\mid a_n$ . Mod $q$ : $a_np^n\equiv0\pmod q$ . Since $\gcd(p,q)=1$ , $q\mid a_n$ .

Conclusion. Any rational root has the stated form.

Corollary. If $P$ is monic ( $a_n=1$ ), any rational root is an integer dividing $a_0$ .

Rational root theorem and irrationality

Statement

Solution

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