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 $n$ .

单词	AlternativeProofThatsqrt2IsIrrational
释义	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 $n$ .
随便看	ell^p ellpXSpace Embedding embedding EmileLemoine Emirp emirp EmmyNoether Emmy Noether EmpiricalDistributionFunction empirical distribution function EmpiricalProofThatSolvingOpposingFacesOfARubiksCubeDoesNotNecessarilySolveTheMiddleLayer empirical proof that solving opposing faces of a Rubik’s cube does not necessarily solve the middle layer EmptyProduct empty product EmptySet empty set EmptySum empty sum EncodingWords encoding words Endpoint endpoint EngelsTheorem Engel’s theorem

alternative proof that 2 is irrational

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