TITLE: How do we come up with the idea in the proof of $x^4+1$ is irreducible over $\Bbb Z$ by tranfer into $(x+1)^4+1$?
QUESTION [6 upvotes]: $x^4+1$ is irreducible over $\Bbb Z$ (hence $\Bbb Q$). The proof I saw is to try transforming it into $(x+1)^4+1=x^4+4x^3+6x^2+4x+2$ and use the Eisenstein criterion with prime $2$. I can understand the proof. However, how do we come up with this idea? How do we know we should try to transfer $x$ into $x+1$, rather than, say $3x-5$? And by the way, I also wonder is there other way to proof the irreducibility?

REPLY [3 votes]: Another way: the polynomial $x^4+1$ doesn't have an integral root, so $x^4+1 \ne (x-a)(x^3+bx^2+cx+d)$. 
we assume that $x^4+1=(x^2+ax+b)(x^2+cx+d)$, then $a+c=0, ac+b+d=0, ad+bc=0, bd=1$, and so $x^4+1=(x^2-\sqrt2 x+1)(x^2+\sqrt2 x+1)$. i.e.  $x^4+1$ is irreducible over $\mathbb Z$.