# 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 |
---|---|

Canonical name | AlternativeProofThatsqrt2IsIrrational |

Date of creation | 2013-03-22 16:55:15 |

Last modified on | 2013-03-22 16:55:15 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 8 |

Author | Wkbj79 (1863) |

Entry type | Proof |

Classification | msc 11J72 |

Classification | msc 12E05 |

Classification | msc 11J82 |

Classification | msc 13A05 |

Related topic | Irrational |

Related topic | EisensteinCriterion |

Related topic | GausssLemmaII |