Exercise 99Number theory › IrrationalityL1Added 3 Nov 2026

A general irrationality criterion

#criterion#irrationality#polynomial

Statement

Let $P(X)=a_nX^n+\cdots+a_0\in\mathbb{Z}[X]$ be monic (i.e. $a_n=1$ ). Show that any real root of $P$ is irrational or an integer.

Hint

Rational root theorem: rational roots are integers dividing $a_0$ .

Show solutionHide solution

Rational root theorem. Any rational root is of the form $p/q$ with $p\mid a_0$ , $q\mid a_n$ . Since $a_n=1$ , $q=1$ , so the root is an integer.

Conclusion. Any non-integer real root of $P$ is irrational.

Application. $X^3-2$ is monic, so $\sqrt[3]{2}$ is irrational; $X^2-2$ is monic, so $\sqrt2$ is irrational.