Today I Learned

Some of the things I've learned every day since Oct 10, 2016

73: Eisenstein’s Criterion

Eisenstein’s criterion provide a sufficient (but not necessary) set of criterion for a polynomial with integer coefficients

$P(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0$

to be irreducible over the rationals. The criterion are that there exists a prime $p$ such that

1. $p$ divides $a_i$ where $i \neq n$
2. $p$ doesn’t divide $a_n$
3. $p^2$ doesn’t divide $a_0$

If all these conditions are true, then $P(x)$ can’t be reduced over the rationals.