Eisenstein's criterion

Posted 2021-02-14 yuanjiangw

 

https://en.wikipedia.org/wiki/Eisenstein%27s_criterion

 

In mathematics, Eisenstein‘s criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorizable into the product of non-constant polynomials with rational coefficients.

This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial.

This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it.^[1][2]