How can I prove that this quintic is irreducible in $\mathbb{Q}$?

Question

How can I prove that $x^5 + 7x^4 + 2x^3 + 6x^2 - x + 8$ is irreducible in $\mathbb{Q}[x]$? I can't use the Eisenstein's criterion and I tried to put this polynomial in $\mathbb{F}_3$ and $\mathbb{F}_5$.

Can you give me some advice please?

Answer 1

Reducing your polynomial modulo $2$ shows that it factors as $$x^5+x^4+x\equiv x(x^4+x^3+1)\pmod{2}^.$$ You can check that $x^4+x^3+1$ is irreducible in $\Bbb{Z}_2$. So if your polynomial were reducible in $\Bbb{Q}$, then it should have an irreducible factor of degree $4$. But then it also has a linear factor, and hence a root in $\Bbb{Q}$. By the rational root theorem, this root must be in $$\{-8,-4,-2,-1,1,2,4,8\}.$$ Plugging all these values in shows that they are not roots of your polynomial, so it is not reducible.

Answer 2

HINT: Show that it cannot be factored over $\mathbb{Z}$. From the hint of @ Element118: it's enough to show that it does not have factors of degree $2$. In fact it is irreducible $\mod 3$, but you only need to show it does not have irreducible factors of degree $2$ mod $3$, and this you can do by a short check.