alternative proof that 2 is irrational

Following is a proof that 2 is irrational.

The polynomialPlanetmathPlanetmath x2-2 is irreducible over by Eisenstein’s criterion with p=2. Thus, x2-2 is irreducible over by Gauss’s lemma ( Therefore, x2-2 does not have any roots in . Since 2 is a root of x2-2, it must be irrational.

This method generalizes to show that any number of the form nr is not rational, where r with r>1 and n such that there exists a prime p dividing n with p2 not dividing n.

Title alternative proof that 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