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单词 AlternativeProofThatsqrt2IsIrrational
释义

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 (http://planetmath.org/GausssLemmaII). 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.

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