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Prove that the polynomial is irreducible: f(x)=x^{6}+30x^{5}-15x^{3}+6x-120\in {\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}}\,.

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