# Fields2

Prove that the polynomial is irreducible: $f(x)=x^6+30x^5-15x^3+6x-120\isin \mathbb{Z}[x]\,$
Let the ideal $P=(3)\subset \mathbb{Z}\,$. By Eisenstein's criterion, all coefficients except the first are elements in $P\,$, and the constant term is not an element of $P^2\,$. Therefore $f(x)\,$ is irreducible over $\mathbb{Z}\,$.