# alternative proof that $\sqrt{2}$ is irrational

Following is a proof that $\sqrt{2}$ is irrational.

The polynomial $x^{2}-2$ is irreducible over $\mathbb{Z}$ by Eisenstein’s criterion with $p=2$. Thus, $x^{2}-2$ is irreducible over $\mathbb{Q}$ by Gauss’s lemma (http://planetmath.org/GausssLemmaII). Therefore, $x^{2}-2$ does not have any roots in $\mathbb{Q}$. Since $\sqrt{2}$ is a root of $x^{2}-2$, it must be irrational.

This method generalizes to show that any number of the form $\sqrt[r]{n}$ is not rational, where $r\in\mathbb{Z}$ with $r>1$ and $n\in\mathbb{Z}$ such that there exists a prime $p$ dividing $n$ with $p^{2}$ not dividing $n$.

Title alternative proof that $\sqrt{2}$ is irrational AlternativeProofThatsqrt2IsIrrational 2013-03-22 16:55:15 2013-03-22 16:55:15 Wkbj79 (1863) Wkbj79 (1863) 8 Wkbj79 (1863) Proof msc 11J72 msc 12E05 msc 11J82 msc 13A05 Irrational EisensteinCriterion GausssLemmaII