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Prove that the polynomial is irreducible: x^{4}+4x^{3}+6x^{2}+2x+1\in {\mathbb  {Z}}[x]\,

The image of f(x)\, in {\mathbb  {Z}}/3{\mathbb  {Z}}[x]\, is f'(x)=x^{4}+x^{3}+2x+1\,.

There are no linear factors: f'(0)=1,f'(1)=2,f'(2)=2\,.

Checking division by all quadratic irreducibles will show that f'(x)\, is irreducible in {\mathbb  {Z}}/3{\mathbb  {Z}}[x]\, and since f(x)\, is primitive, f(x)\, is irreducible over {\mathbb  {Z}}\,.

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