How to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers?

Question

Which criterion (test) one can use in order to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers ?

Neither of Eisenstein's criterion and Cohn's criterion cannot be applied on this polynomial. I know that one can use factor command in Wolfram Alpha and show that polinomial is irreducible but that isn't point of this question.

Answer 1

Let your polynomial be $f$. Clearly it has no linear factor, since it has no root in $\mathbb{Z}$. Hence if it factors, it factors as the product of two irreducible quadratics $f_1, f_2$.

Now looking mod $2$ we get a factorization $f=x^4+x^3+x^2+x+1=f_1f_2$ in $\mathbb{F}_2[x]$. Now since $f$ has no root mod $2$, $f_1$ and $f_2$ are also irreducible quadratics in $\mathbb{F}_2[x]$. But the only irreducible quadratic in $\mathbb{F}_2[x]$ is $x^2+x+1$. This would imply $x^4+x^3+x^2+x+1 = (x^2+x+1)^2$ in $\mathbb{F}_2[x]$, which is false.

Answer 2

Hint: If a quartic is reducible it has either a linear factor or a quadratic factor. It is easy to check that your polynomial above has no linear solutions, and you can work out a contradiction if you assume that it can be factored into the product of two (monic) quadratics.