Prove that the following polynomial is irreducible over $\mathbb{Z}$:
$$f(x) = x^8-x^7+x^5-x^4+x^3-x+1$$
My attempt: one can see that $f(x)=(x^4+x^3+1)(x^4+x+1) $ over $\mathbb{F_2}$ where these 2 polynomials are irreducible, so if $f$ is factorizable, then
$$f(x)=g(x)h(x)$$ for $g,h\in\mathbb{Z[x]}$ - of degree 4.
I tried comparing coefficients, but it turned out to be extremely tedious. Another approach is to plug some fancy numbers into $f(x)$: I found that $f(x)-1 $ is divisible by $x\cdot(x+1)\cdot(x-1)\cdot(x^2+1)$, so $g(n)=1$ or $-1 $ for $n=-1,0,1$, but there are also too many cases. Now I have no idea how tackle it.