alternative proof that is irrational
Following is a proof that is irrational.
The polynomial is irreducible over by Eisenstein’s criterion with . Thus, is irreducible over by Gauss’s lemma (http://planetmath.org/GausssLemmaII). Therefore, does not have any roots in . Since is a root of , it must be irrational.
This method generalizes to show that any number of the form is not rational, where with and such that there exists a prime dividing with not dividing .
|Title||alternative proof that is irrational|
|Date of creation||2013-03-22 16:55:15|
|Last modified on||2013-03-22 16:55:15|
|Last modified by||Wkbj79 (1863)|